THE    INFLUENCE    OF    MOLECULAR 
CONSTITUTION 


UPON    THE 


INTERNAL  FRICTION  OF  GASES. 


BY 

FREDERICK  MALL1NG  PEDERSEN,  E.E.,  Sc.  D. 

Instructor   in   Mathematics    in    the   College   of   the   City  of  New   York. 


SUBMITTED  IX  PARTIAL    FULFILLMENT  OF  THE  REQUIREMENTS 

FOR  THE  DEGREE  OF  DOCTOR  OF  SCIENCE  IN  THE 

FACULTY  OF  SCIENCE,  NEW  YORK  UNIVERSITY, 

APRIL,   1905. 


Hew 

D.  VAN  XOSTRAND  COMPANY, 

23  MURRAY  AND  27  WARREN  STREETS. 

1906. 


THE    INFLUENCE    OF    MOLECULAR 
CONSTITUTION 


UPON    THE 


INTERNAL  FRICTION  OF  GASES. 


BV 


FREDERICK  MALL1NG  PEDERSEN,  E.E.,  Sc.  D. 

Instructor    in    Mathematics    in    the   College   of   the   City  of  Xeic   York. 


SUBMITTED   IX   PARTIAL    FULFILLMENT  OF  THE   REQUIREMENTS 

FOR  THE    DEGREE   OF   DOCTOR  OF   SCIENCE   IN  THE 

FACULTY  OF  SCIENCE,  NEW  YORK  UNIVERSITY, 

APRIL,  1905. 


D.  VAN  NOSTRAND  COMPANY, 

23  MURRAY  AND  27  WARREN  STREETS. 

1906. 


Ill 


TABLE   OF  CONTENTS. 

PAGE 

INTRODUCTION 1—3 

HISTORICAL  REVIEW 3-34 

Baily 7 

Barus 29 

Bernoulli 3 

Bessel 6 

Bestelmeyer 33 

Boltzmann 28 

Braun  &  Kurz - 24 

Breitenbach 32 

Chezy 6 

Clausius 9 

Couette 28 

Coulomb 5 

Couplet 4 

Crookes 23 

De  Keen 28 

Du  Buat 4 

Euler 4 

Eytehvein 5 

Gerstner 5 

Girard 6 

Girault 10 

Graham 8 

Green 7 

Grossman 24 

Guthrie 17 

Hagen 7 

Helmholtz  &  Pietrowski 10 

Hoffmann 25 

Holman 17 

Houdaille 31 

Jaeger 31 

Job 32 

Klemencic 24 

Kirchhoff 17 

Koch 24 

Koenig 27 

Kundt  &  Warburg 15 


239372 


PAGE 

Kleint 33 

Lampe 28 

Lampel 28 

Lang 13 

Ludwig  &  Stefan 9 

Marian 5 

Margules. 23 

Markowski 33 

Maxwell 9 

Mathieu 10 

Meyer,  L 12,  19,  22 

Meyer,  L.  &  Schumann *. 20 

Meyer,  O.  E 11,  13,  14,  28 

Meyer  &  Sptingmuhl 14 

Naumann 12 

Navier 6 

Noyes  &  Goodwin 30 

Newton 3 

Obermayer 16 

Ortloff 30 

Perot  &  Fabry 31 

Poiseuille 8 

Poisson 7 

Prony 5 

Puluj 15,  17,  18 

Rayleigh 32 

Reynolds,  F.  G 33 

Reynolds,  O 25 

Sabine. . .' 6 

Schneebeli 27 

Schultze 32 

Schumann 26 

Stefan 18 

Steudel 21 

Stewart  &  Tait 11 

Stokes 9 

Sutherland 29 

Tomlinson 27 

Warburg 16 

Warburg  &  Babo 24 

Wiedemann,  E ^ 17 

METHOD  EMPLOYED  IN  EXPERIMENTAL  INVESTIGATION 34 

GENERAL  DESCRIPTION  OF  APPARATUS 34 

APPARATUS  No.  1 38 

Internal  Friction  of  Air,  Apparatus   No.  1 39 

APPARATUS  No.  2 40 

Internal  Friction  of  Air,  Apparatus  No.  2  and  No.  3 42 

PREPARATION  OF  ETHERS , 42 

Internal  Friction  of  Ethers,  Apparatus  No.    2 43 


PAGE 

APPARATUS  No.  3 45 

Internal  Friction  of  Ethers,    Apparatus  No.  3 46 

MOLECULAR  VOLUMES 48 

MOLECULAR  MEAN  SPEEDS,  FREE  PATHS,  AND  COLLISION  FREQUEN- 
CIES  50,  51 

COMPARISON  WITH  THE  RESULTS  OF  OTHERS 51 

SUMMARY  OF  RESULTS 52 

BIBLIOGRAPHY . .  54-59 


THE  INFLUENCE  OF  MOLECULAR  CONSTITUTION 
UPON  THE  INTERNAL  FRICTION  OF  GASES. 


BY   FREDERICK  M.   PEDERSEN. 


INTRODUCTION. 

The  effect  of  molecular  structure  upon  the  physical  prop- 
erties of  matter  has  interested  many  scientists,  but  much 
about  this  question  yet  remains  to  be  learned.  In  the  hope 
of  making  a  contribution  to  the  subject  the  following  investi- 
gation was  undertaken. 

A  careful  historical  research  reveals  the  fact  that  the  in- 
ternal friction  of  most  of  the  isomeric  ether  gases  that  I  have 
employed  has  never  before  been  determined.  Nevertheless, 
my  object  has  been,  not  so  much  to  determine  with  great 
accuracy  the  absolute  values  of  these  frictions,  as  to  obtain 
by  an  extremely  sensitive  method  accurate  comparative  re- 
sults, in  the  hope  of  proving  that  difference  in  molecular  con- 
stitution is  accompanied  by  measurable  difference  in  friction, 
and  therefore  by  a  difference  in  the  size  of  the  molecule. 

By  the  internal  friction  or  viscosity  of  a  gas,  is  meant  the 
friction  between  two  adjacent  layers  of  the  gas,  which  are 
moving  in  parallel  directions  but  not  with  the  same  velocity. 
It  is  proportional  to  the  relative  velocity  and  the  area  of  con- 
tact of  the  two  layers.  The  coefficient  of  internal  friction 
is  the  constant  factor,  depending  upon  the  kind  of  gas  under 
consideration,  which,  when  multiplied  by  the  difference  in 
velocity,  gives  the  friction  per  square  unit  of  surface  of  contact 
of  the  two  layers. 

Although  there  is  no  sensible  cohesion  between  the  par- 
ticles of  a  gas,  as  there  is  in  liquids  and  solids,  the  tendency 
of  a  moving  layer  of  gas  to  impart  its  own  velocity  to  a  layer 
in  contact  with  it,  has  been  satisfactorily  explained  according 
to  the  kinetic  theory  of  gases.  As  is  well  known,  all  the  mole- 


cules  of  a  gas  are  supposed  to  be  constantly  in  rectilinear 
motion,  rebounding  in  straight  lines  with  undiminished  ve- 
locity, after  collision  with  one  another  or  a  solid  wall.  In 
virtue  of  its  mass  and  velocity  each  molecule  possesses  mo- 
mentum. When  one  layer  of  gas  is  moving  over  another, 
there  is  a  constant  passage  of  molecules  from  each  layer  into 
the  other.  There  is  then  an  interchange  of  momentum,  the 
result  being  a  tendency  to  equalize  the  velocities  of  the  two 
layers.  We  see  thus  that  the  effect  of  friction  is  produced 
between  two  layers  of  gas  which  are  not  moving  with  the  same 
velocity  in  the  same  direction. 

It   has  been   demonstrated1  that   the   coefficient   of  friction 
of  a  gas, 

m  G 


47TS2 

where  m  =  the  mass  of  a  molecule,  G  =  the  mean  value  of 
the  velocity  as  deduced  from  the  mean  kinetic  energy,  and 
5  =  the  distance  between  the  centres  of  two  molecules  at 
impact,  or  the  diameter  of  the  sphere  of  action.  This  formula 
brings  out  the  remarkable  and  very  important  law,  that  the 
viscosity  of  a  gas  is  independent  of  the  pressure  at  a  constant 
temperature.  It  also  shows  that  the  smaller  the  sphere  of 
action,  the  greater  will  be  the  coefficient  of  viscosity.  We 
also  see  that  the  viscosity  varies  directly  as  the  rectilinear 
velocity  of  the  molecules,  i.e.,  as  the  square  root  of  the  kinetic 
energy,  and  therefore  as  the  square  root  of  the  absolute  tem- 
perature, since  the  last  two  are  directly  proportional  to  each 
other.  The  first  named  law  has  been  found  true  experimentally 
for  all  ordinary  pressures,  but  the  last  law  connecting  the 
viscosity  and  absolute  temperature  has  not  been  substantiated 
by  experiment.  This  disagreement  between  theory  and  prac- 
tice I  will  leave  for  a  fuller  discussion  later,  merely  .noting  in 
passing  that  Maxwell2  has  shown  that  the  assumption  in  regard 
to  the  nature  of  the  impact  between  two  molecules  determines 
the  relation  between  the  coefficient  of  friction  and  the  absolute 
temperature. 

That  the  coefficient  of  friction  of  a  gas  is  independent  of 
the  pressure  seems  at  first  inconceivable,  but  it  can  probably 

1.  O.  E.  Meyer's  Kinetic  Theory  of  Gases,  1899,  p.  179. 

2.  Phil.  Mag.,   1868  (4),  Vol.  35,  p.  211. 


be  better  understood  by  a  consideration  of  another  formula 
which  has  been  deduced  from  the  kinetic  theory,  viz.:  T)  =  1/3 
d  G  Ll  where  d  is  the  density,  G  is  square  root  of  the  mean 
square  of  velocity,  and  L  is  the  mean  free  path  of  a  molecule. 
It  can  be  shown  that  L  and  d  are  inversely  proportional  to 
each  other  so  that  their  product  is  constant,  an,d  therefore  77 
is  thus  independent  of  a  change  in  density. 

Before  describing  my  own  experimental  investigations  I 
will  give  a  historical  review  of  the  work  done  by  previous 
experimenters  in  the  subject  of  the  viscosity  of  gases.  Methods 
of  research  in  this  line  seem  to  have  developed  out  of  or  along 
with  investigations  in  the  subject  of  the  internal  friction  of 
liquids.  I  shall  therefore  frequently  have  occasion  to  refer 
to  this  last  named  subject,  but  I  shall  make  no  attempt  what- 
ever to  follow  out  its  complete  development. 
HISTORICAL  REVIEW. 

O.  E.  Meyer2  points  out  that  Newton3  made  the  first  at- 
tempt at  a  theory  of  the  friction  of  fluids.  His  fundamental 
hypothesis  was  that  the  friction  between  two  adjacent  layers 
of  a  fluid  is  proportional  to  the  difference  of  velocity,  the  area 
of  contact  and  independent  of  the  pressure.  This  hypothesis 
has  several  times  since  been  enunciated  by  others,  independently 
of  Newton,  and  used  as  a  basis  for  a  theory  of  fluid  friction. 
Unfortunately  Newton's  mathematical  development  of  the 
theory  from  this  assumption  is  erroneous.  His  experiments, 
like  those  of  many  early  investigators,  seem  to  have  been 
confined  to  observations  of  pendulums.  It  is  interesting  to 
note  that  on  searching  for  the  resistance  which  air  opposes 
to  the  movement  of  a  globe,  swinging  in  very  small  arcs,  he  used 
a  formula  composed  of  three  terms:  one  containing  the  square, 
the  second  the  3/2  power  and  the  third  the  first  power  of  the 
velocity.4  Later,  1719,  in  calculating  the  resistance  which  is 
offered  to  spheres  falling  slowly  in  air  or  water,  he  reduced  the 
formula  to  two  terms:  one  varying  as  the  square  of  the  velocity, 
and  the  other  constant. 

Bernoulli5  pointed   out   Newton's  error  in   his   paper  before 


1     Meyer's  Kinetic  Theory  of  Gases,   1899,  p.    1 78. 

2.  Crelle's  Journal  fur  Mathematik,  1861,  Bd.  59,  p.  22'J 

3.  Philosophiae  naturalis  principiamathematica,  1687,  Lib.  II.,  Sec.  IX. 

4.  IVincipia  Lib.  II.,  Prop.  XL. 

5.  Opera  Ornnia.      Lausanne  et  Genevae   1742.    Tornus  III.    Xouvelles 
pense>s  snr  le  systeme  de  M.  Descartes,  XIX. -XXIII. 


the  French  Academy  in  1730,  explaining  the  orbits  of  the  planets 
and  the  precession  of  the  equinoxes,  according  to  the  Cartesian 
vortex  hypothesis.  On  submitting  to  calculation  the  experi- 
ments with  pendulums  made  by  Newton,  Bernoulli  supposes 
only  two  terms  to  represent  the  resistance;  one  varying  as  the 
square  of  the  velocity,  the  other  constant.  He  points  out 
that  the  theory  does  not  agree  with  the  experiments,  but 
states  that  we  cannot  conclude  anything  from  that  because 
of  the  difficulty  and  delicacy  of  the  observations  necessary.1 

Couplet,2  in  1732,  called  attention  to  the  fact  that  the  rules 
then  in  use  for  determining  the  flow  of  water  through  pipes 
were  utterly  useless,  as  the  error  amounted  to  nearly  100%. 
He  made  some  experiments  on  a  large  scale  at  Versailles,  and 
found  that  the  water  delivered  was  frequently  1/20  or  1/30 
what  was  promised  by  his  calculations.  Nothing  was  known 
about  internal  friction,  the  effect  of  bends,  diameter  of  pipe, 
etc.  Couplet  apparently  made  no  attempt  to  discover  the 
true  laws  of  "flow  and  seemed  to  scarcely  believe  in  the  possi- 
bility of  discovering  them. 

Euler3  in  1756  fell  into  the  error  of  supposing  that  the  fric- 
tion of  a  liquid  is  independent  of  the  velocity  and  propor- 
tional to  the  hydrostatic  pressure. 

Du  Buat4  30  years  later  made  a  series  of  quite  elaborate 
experiments  with  pendulums  consisting  of  spheres  of  lead, 
wood  and  paper,  which  he  oscillated  in  water  and  air.  He 
noted  that  similar  laws  seemed  to  hold  in  both  media.  He 
noticed  that  both  liquid  and  air  were  dragged  along  by  the 
spheres.  He  found  the  resistance  to  the  motion  of  the  spheres 
was  proportional  to  their  surfaces,  and  that  the  resistances 
of  air  and  water  are  as  their  densities.  He  calculated  the 
resistance  of  the  air  to  Newton's  falling  spheres,  and  found 
that  up  to  a  velocity  of  23  feet  per  second  the  resistance  is  as 
the  square  of  the  velocity.  At  higher  velocities  he  showed 
it  was  greater  owing  to  a  constant  vacuum  back  of  the  ball. 
He  pointed  out  a  double  correction  which  must  be  applied 
to  a  pendulum  in  reducing  the  results  to  a  vacuum.  1°  for 
its  apparent  loss  in  weight  owing  to  the  buoyancy  of  the  air. 

1.  Me"moires  de  Petersbourg.     Tome  III.  and  V. 

2.  M&noires  de  1'Academie,   1732. 

3.  Tentomen  theoriae  de  frictione  fluidorum.    Novi  Petropolitani,  Tomus 
VI.,  1756;  7  Pag.  338. 

4.  Principes  d'Hydrauliques,  1786,  Vol.  2,  Part  3;  Sec.  2,  p.  279. 


and  2°  for  the  increase  in  the  moment  of  inertia  due  to  the 
air  clinging  to  it.  A  full  discussion  and  analysis  of  his  work 
with  pendulums  was  given  by  Professor  Stokes1  in  1850. 

Some  years  previous  to  Du  Buat's  work  Mairan2  made  quite 
an  important  advance  in  the  experimental  study  of  pendulums 
by  the  invention  of  the  method  of  coincident  observations 
of  two  pendulums,  a  method  made  use  of  by  many  later  in- 
vestigators. 

Gerstner,3  through  experiments  made  in  Prague  in  1796, 
discovered  the  very  great  effect  which  temperature  has  upon 
the  mobility  of  water.  He  found  that  in  many  cases  a  rise 
of  20  or  30  degrees  in  temperature  doubled  the  quantity  of 
liquid  delivered  through  narrow  tubes,  thus  showing  a  very 
marked  decrease  in  the  internal  friction  of  the  water. 

Coulomb4  in  1801  published  an  account  of  experiments 
undertaken  to  determine  the  cohesion  of  fluids  and  the  laws 
of  their  resistance  when  in  very  slow  motion.  He  claims 
that  the  expression  for  the  resistance  of  a  fluid  has  two  terms, 
one  proportional  to  the  square  of  the  velocity,  and  the  other 
to  its  first  power,  and  that  if  there  is  a  constant  term,  it  is 
so  small  in  all  fluids  of  small  cohesion,  that  it  is  almost  im- 
possible to  appreciate  it.  Instead  of  using  the  pendulum 
method,  he  noted  the  diminution  in  amplitude  of  oscillation 
of  horizontal  disks,  oscillating  in  their  own  plane  under  the 
torsion  of  a  brass  wire.  He  was  the  first  one  to  adopt  this 
important  method  which  has  since  been  employed  by  many 
others. 

Both  Prony5  in  1804,  and  Eytelwein8  ten  years  later, 
developed  theories  of  the  flow  of  water  through  pipes,  but 
with  the  temperature  left  entirely  out  of  consideration.  Prony 
found  the  loss  of  head  per  unit  length  is  very  nearly  proportional 
to  the  square  of  the  mean  velocity  of  the  water.  He  proposed 
the  formula:  1/4DJ  =av+ftvi  where  D  =  diameter  of  the 
pipe,  /  =  the  fall  per  metre,  v  =  velocity,  a  =0.0000173314, 
ft  =  0.0000348259. 

1.  Cambridge  Phil.  Trans.,  1856,  Vol.  9,  Part  2,  p.  8. 

2.  Historic  de  1'Acad.  de  Paris,  1735,  p.  166. 

3.  Gilbert's  Annalen,  Bd.  5,  1800,  p.   160.     Neu.Abh.derkon.B6hm. 
Gesell.  der  Wiss.,  Bd.  3,  Prag.  1798. 

4.  Mem.  de  1'Institut  Nat.  des  Sciences  et  Arts.  Annee  9,  Tome  III., 
p.  246. 

5.  Recherches   Physico-mathem.    sur    la  The*orie  des    Eaux  Courantes, 
Paris,  1804. 

6.  Abh.  d.  Berl.  Akad.,  1814  and  1815. 


Girard1  in  1815  investigated  the  flow  of  liquids  through 
capillary  tubes  of  copper  with  special  reference  to  the  effect 
of  temperature.  He  found  that  at  86°  C.  four  times  as  much 
water  was  delivered  as  at  0°  C.  He  noted  that  when  the  cap- 
illary reached  a  certain  length  the  term  which  is  proportional 
to  the  square  of  the  velocity  disappears  from  the  formula  for 
the  uniform  motion  of  liquids.  In  fine  tubes  he  found  the  loss 
of  head  per  unit  length  very  nearly  proportional  to  the  ve- 
locity of  flow.  He  refers  to  the  work  of  Chezy2  in  1775  as 
being  the  first  investigation  of  the  flow  of  water  in  aqueducts. 

In  1823  Navier3  deduced  the  differential  equations  of  the 
motion  of  a  viscous  medium  from  considerations  which  are 
analogous  to  those  he  employed  for  the  derivation  of  the  differ- 
ential equations  of  the  phenomena  of  elasticity.  He  treats 
of  the  movement  of  a  fluid  in  a  straight  pipe  of  circular  cross 
section.  In  a  later  paper  he  discusses  the  discharge  of  air 
from  pipes  and  orifices,  and  endeavors  to  take  into  account 
the  effect  of  bends.  The  velocity  of  flow  seems  to  have  been 
his  chief  concern.  The  smallest  pipe  he  used  had  a  diameter 
of  1.579  cm. 

In  1826  Bessel4  published  the  result  of  his  classical  investi- 
gation of  the  length  of  the  simple  second's  pendulum.  His 
method  consisted  in  the  comparison  of  two  pendulums  whose 
difference  in  length  was  exactly  one  toise  or  1.92  metres.  He 
took  account  of  the  effect  of  the  resistance  of  the  air  and 
showed  that  the  old  correction  for  reduction  to  a  vacuum 
was  in  error  and  determined  its  correction  factor,  viz. :  k  =  1.946 
for  a  sphere  about  5  cm.  in  diameter. 

Sabine5  made  the  interesting  experiment  in  1829  of  swinging 
a  pendulum  in  air  under  a  pressure  of  about  half  an  atmosphere, 
and  in  hydrogen  at  atmospheric  pressure,  and  found  that 
hydrogen  had  a  much  greater  resistance  in  proportion  to  its 
density  than  had  air.  He  correctly  ascribed  the  cause  to  an 
inherent  property  of  elastic  fluids,  independent  of  their  density, 
analogous  to  that  of  viscosity  of  liquids.  He  also  investigated 
the  correction  for  reduction  to  a  vacuum.  His  pendulum 


1.  Me"m.  de  1'Institut,  Classe  Sc.  and  Math.,  1813-15,  p.  248;  1816,  p.187. 

2.  Mdm.  manuscript  de  1'Ecole  des  Pons  et  Chausse"es,  1775. 

3.  Me"m.  de  1'Acad.   Roy.  des  Sciences,   1823,  Tome  6,   p.   389,    1830, 
Tome  9,  p.  311. 

4.  Abb.    Herl.  Akad.  Math.  Klasse,   1826,  p.   1. 

r>    Phil.  Trans.,  1829,  p.  207  and  331;  1831,  p.  470. 


made  10.36  more  oscillations  in  a  given  time  in  a  vacuum 
than  in  air,  whereas  calculations  showed  that  it  ought  to  make 
only  6.26  more,  hence  the  correction  for  reduction  to  a  vacuum 
was  in  error. 

In  1829JLPoisson1  derived,  by  a  method  similar  to  Navier's, 
the  differential  equations  of  equilibrium  and  motion  of  fluids. 
Two  years  later  he  wrote  a  paper  on  the  motion  of  a  pendulum 
in  a  resisting  medium,  in  which  he  discusses  the  correction 
for  reduction  to  a  vacuum  and  infinitely  small  arcs,  and  reviews 
the  work  of  Du  Buat,  Bessel  and  Sabine. 

In  1832  Baily2  made  a  very  thorough  and  systematic  study 
of  the  correction  of  a  pendulum  for  the  reduction  to  a  vacuum. 
He  oscillated  86  different  pendulums  with  bobs  of  various  mate- 
rials, sizes  and  shapes  in  air  and  then  in  a  vacuum.  In  the 
case  of  a  spherical  bob  two  inches  in  diameter  he  found  the 
old  correction  should  be  multiplied  by  1.748  which  agrees 
quite  well  with  Bessel's  value  of  k  but  not  so  well  with  Du 
Buat's,  1.585.  He  found  the  correction  varies  with  the  shape 
but  not  with  the  specific  gravity  of  the  pendulum. 

Green,3  continuing  researches  on  the  vibration  of  pendulums 
in  fluid  media,  claimed  that  in  the  case  of  a  sphere  the  density 
of  the  pendulum  should,  in  the  calculations,  be  augmented 
by  one  half  the  density  of  the  surrounding  fluid,  while  its 
weight  is  diminished  by  the  weight  of  the  volume  of  the  fluid 
it  displaces. 

Hagen,4  returning  in  1839  to  the  investigation  of  the  flow 
of  water  in  small  cylindrical  tubes,  tried  to  improve  on  Prony's 
and  Eytelwein's  equations  by  taking  account  of  the  tempera- 
ture. He  criticized  Du  Buat  for  not  recording  the  temperature 
in  any  of  his  experiments  in  spite  of  the  fact  that  he  knew 
warm  water  is  more  fluid  than  cold  water.  Hagen  found  the 
rapidity  of  flow  increases  with  the  temperature  to  a  maximum, 
then  diminishes  for  a  rise  of  10  or  20  degrees,  and  then  begins 
to  increase  again.  With  large  tubes  and  high  speeds  both 
turning  points  fell  below  the  freezing  point.  With  very  narrow 
tubes  and  low  speeds  they  both  fell  above  the  boiling  point. 


1.  Journal  de  1'Ecole  Polytech.,  1831,  Tome  13,  p.   139.     Connaissance 
des  Terns,  1834,  Appendix.     Mem.  de  1'Acad.,  1832,  Tome  2,  p.  521. 

2.  Phil.  Trans.,   1832,  Part  1,   p.  399. 

3.  Trans.  Royal  Soc.  Edin.,  Vol.  13,   1836,  p.  54. 

4.  Pogg.  Ann.,  1839,  Bd.  46,  p.  423.     Abh.  d.  Berl.  Akad.,  1854,  p.    17. 


In  the  following  year  Poiseuille1  published  the  results  of 
his  epoch-making  experimental  research  on  the  movement  of 
liquids  in  tubes  of  very  small  diameter.  He  used  .glass  capil- 
laries from  0.013  mm.  to  0.652  mm.  in  diameter,  and  from 
2  to  800  mm.  long.  They  were  slightly  elliptical  in  cross  sec- 
tion, so  he  took  the  mean  diameter.  The  liquid,  water  or  alcohol, 
was  contained  in  a  bladder  and  pressed  out  through  the  cap- 
illary. He  investigated  the  effect  on  the  quantity  of  liquid 
discharged,  of  the  pressure,  length  of  tube  and  the  tempera- 
ture. He  found  the  volume  of  liquid  discharged  varies  directly 
.as  the  pressure,  inversely  as  the  length  of  tube,  directly  as 
the  fourth  power  of  the  diameter,  and  with  the  temperature 
according  to  an  empirical  law.  He  found  the  value  of  a  certain 
factor  to  be  constant  for  all  sizes  of  tubes.  This  factor  for 
water  was  1836.7  at  0°  C.  and  2495.9  at  10°  C. 

What  Poiseuille  had  done  for  liquids,  Graham2  did  for  gases 
a  few  years  later.  He  first  investigated  the  effusion  of  gases 
into  a  vacuum  through  a  very  fine  aperture  in  a  thin  plate, 
and  found  the  velocity  of  effusion  varies  inversely  as  the 
square  root  of  the  density  of  the  gas.  He  determined  the 
effusion  coefficients  of  a  number  of  gases  referred  to  air  and 
oxygen  as  unity.  On  the  other  hand,  on  causing  gases  to 
transpire  through  capillary  tubes  which  were  long  in  com- 
parison with  their  diameter,  he  discovered  the  transpiration 
coefficients  to  be  independent  of  the  density  of  the  gases.  He 
either  let  the  gas  flow  from  a  gas  holder  through  the  capillary 
into  a  partial  or  complete  vacuum,  or  from  a  com- 
pression cylinder  into  the  air.  In  both  cases  the 
pressures  were  not  constant  during  the  experiment. 
He  found  the  velocities  of  transpiration  to  vary  directly 
as  the  pressure,  and  inversely  as  the  length  of  tube. 
A  rise  in  temperature  he  found  decreased  the  velocity. 
The  transpiration  rate  was  independent  of  the  material  of  the 
capillaries,  which  were  of  glass  or  copper.  The  presence  of 
moisture  in  the  air  had  but  a  small  effect  upon  its  transpiration. 
The  velocities  with  which  the  different  gases  passed  through 


1.  Soc.   Philomath,   1838,  p.  77,  Comptes  Rendus,   1840,  Vol.  II.,  pp. 
961,  1041.      1841,  Vol.  12,  p.  112;  1842,  Vol.  15,  p.  1167.     Ann.  de  Chim. 
et  de  Phys.      1843  (3),  Vol.  7,  p.  50.     Me"m.  de  Savans  etrangers,  1846, 
Vol.  9,  p.  433. 

2.  Phil.  Trans.,  1846,  Vol    136,  p.  573.     Phil.  Trans.,  1849,  Vol.     139, 
p.  349. 


9 

the  various  capillaries,  bore  a  constant  relation  to  each  other. 
The  compounds  of  methyl  had  a  less  velocity  than  the  corre- 
sponding compounds  of  ethyl,  but  a  constant  relation  appeared 
between  them. 

In  1849  Stokes1  published  an  elaborate  methematical  treat- 
ment of  the  theories  of  the  internal  friction  of  fluids  in  motion, 
and  derived  the  same  complete  equations  of  motion  as  Navier 
and  Poisson  by  a  different  method.  The  following  year  he 
read  a  paper  "  On  the  Effect  of  the  Internal  Friction  of  Fluids 
on  the  Motion  of  Pendulums  "  in  which  he  gave  a  very  full 
discussion  of  the  work  of  Du  Buat,  Bessel  and  Baily  already 
referred  to.  He  was  well  aware  of  the  fact  that  the  air  sticks 
fast  to  a  pendulum,  which  therefore  in  its  oscillations  causes 
a  friction  of  air  on  air.  He  was  in  error,  however,  in  thinking 
that  what  we  now  know  as  the  coefficient  of  friction  was  de- 
pendent upon  the  density.  From  his  own  experiments  and 
those  of  others,  he  deduced  values  of  what  he  calls  the  "  index 
of  friction,"  which  is  our  coefficient  divided  by  the  density. 
He  found  his  results  agreed  well  with  Baily's  but  not  with 
Bessel's.  Baily's  experiments  show  7?  for  air  to  be  =  .000104 
in  c.g.s.  units.  Stokes  also  deduced  the  index  of  friction  of 
water  from  Coulomb's  observations  on  the  decrement  of  the 
arc  of  rotary  oscillation  of  horizontal  disks. 

Ludwig  and  Stefan2  in  1858  called  attention  to  the  formation 
of  eddies  when  a  tube  discharges  liquid  into  one  of  larger  bore, 
thus  causing  an  increase  in  resistance.  A  similar  action  was 
later  shown  to  take  place  in  a  gas  under  the  same  circumstances. 

The  same  year  Clausius3  published  his  important  paper  on 
the  kinetic  theory  of  gases  which  did  so  much  to  establish  that 
theory. 

In  1860  Maxwell4  entered  this  field  of  investigation  and 
made  some  important  contributions  to  the  kinetic  theory  of 
gases,  proving  mathematically  that  T?  must  be  independent 
of  the  density.  Six  years  later  appeared  his  Bakerian 
Lecture  in  which  he  gave  an  account  of  his  experiments  on  the 


1.  Trans.  Camb.  Phil.  Society,  1849  (3),  Vol.  8,  p.  287.     Trans.  Camb. 
Phil.  Society,  1856,  Vol.  9,  p.  8.      Phil.  Mag.,  1851  (4),  Vol.  1,  p.  337. 

2.  Sitzber.  Wien.  Akad.,  1858,  Vol.  32,  p.  25. 

3.  Pogg.  Ann.,  1858,  Bd.   105,  p.  239. 

4.  Phil.  Mag.,  1860  (4),  Vol.  19,  p.  31;  1869  (4),  Vol.  35,  p.  209,  211. 
Phil.  Trans.,  1866,  Vol.  150  (1),  p.  249.     Collected  Papers,  Vol.  II.       Proc 
Royal  Society,   1866.  Vol.    15,  p.   14. 


10 

viscosity  of  air  and  other  gases.  His  method  was  a  modifica- 
tion of  Coulomb's,  in  which  he  used  several  oscillating  disks 
with  stationary  plates  between  them.  The  only  uncertainty 
in  his  mathematical  treatment  of  the  theory  of  his  apparatus 
is  due  to  the  action  which  takes  place  at  the  edges  of  the  oscil- 
lating plates  which  is  not  fully  understood.  His  value  for  q 
for  air  at  18°  C.  is  .0002. 

He  found  from  his  experiments  that  the  coefficient  of  fric- 
tion was  directly  proportional  to  the  absolute  temperature 
instead  of  to  the  square  root  of  the  latter.  He  then  modified 
his  kinetic  theory  of  gases  in  order  to  make  the  theory  agree 
with  his  observations,  by  adopting  the  hypothesis  of  a  repulsive 
force  between  the  molecules,  which  varies  inversely  as  the 
fifth  power  of  the  distance  between  them,  this  being  the  only 
law  of  force  which  causes  T?  to  increase  as  the  first  power 
of  the  absolute  temperature.  His  observations  concerning 
the  increase  of  TJ  were  undoubtedly  in  error  as  other  investigators 
do  not  agree  with  him. 

Helmholtz  and  Pietrowski1  in  1860  tried  an  interesting 
modification  of  Coulomb's  method  which  consisted  in  observing 
the  logarithmic  decrement  of  the  rotary  oscillations  of  a  spher- 
ical vessel  containing  the  liquid  under  investigation.  Helm- 
holtz worked  out  the  mathematical  theory  of  this  method  while 
Pietrowski  performed  the  experiments.  With  the  interior 
surface  of  the  vessel  highly  polished  and  gilded  a  slipping  of 
the  fluid  along  the  side  of  the  vessel  was  thought  to  be  observed. 
The  results  are  to  Poiseuille's  as  4:5.  Probable  sources  of 
error  mentioned  are  vibrations  and  changes  of  temperature. 

Girault2  in  his  pendulum  experiments  of  that  year  brought 
out  the  fact  very  clearly  that  most  of  the  resistance  to  a  pen- 
dulum's motion  resides  at  the  bob  and  not  at  the  point  of 
suspension. 

In  1863  Mathieu3  deduced  Poiseuille's  law  for  capillaries 
of  elliptical  cross  section.  Poiseuille  found 

„  Pressure  X  Diameter4 
length 

where  Q  is  the  quantity  of  liquid  discharged  in  a  unit  time 


1.  Wien.  Ber.  Mathtn.  Naturw.,  1860,  Vol.  40,  p.  607 

2.  Mdm.  de  1'Acad.  de  Caen,  1860. 

3.  Compt.  Rend.,  1863,  Tome  57,  p.  320. 


11 

and  K  is  a  constant  depending  upon  the  nature  of  the  liquid. 
Mathieu  supposes  the  velocity  of  the  liquid  in  contact  with 
the  capillary  wall  to  be  zero,  because  K  is  independent  of  the 
material  of  the  capillary,  and  because  of  observations  on  blood 
in  the  capillaries  of  a  frog's  foot.  For  round  capillaries  he  gives 

.  Pressure  R4 

the   above   equation  the   form  Q  =-— — — —. -—where  N  is  a 

8  i\  length 

constant  depending  upon  the  liquid.      For  elliptical  capillaries 

Pa3b* 

Q  =    i  \T  i  t  2  .  L->     where  a  and  b  are  the  semi-major  and  minor 
4  .V  /  (a2  -f  o2) 

axes  of  the  ellipse. 

In  1865  Stewart  and  Tait1  made  some  interesting  observa- 
tions on  the  heating  of  a  disk  rotating  rapidly  in  vacuo.  They 
found  the  heating  independent  of  the  rarefaction  which  seemed 
to  show  the  friction  independent  of  the  density  or  pressure. 

O.  E.  Meyer,2  who  probably  did  more  than  any  other  one 
man  to  advance  our  knowledge  of  the  viscosity  of  gases,  began 
investigations  in  this  subject  about  this  time.  A  few  years 
before  he  had  begun  the  study  of  the  viscosity  of  liquids,  but 
he  now  turned  his  attention  to  gases.  He  first  adopted  the 
oscillation  method  of  Coulomb  and  found  a  value  of  T?  for 
air  at  18°  C.  of  0.00036  which  is  much  too  large.  Later  he 
pursued  his  investigations  with  pendulums.  His  object  was 
to  determine  what  effect  pressure  and  temperature  have  on 
the  coefficient  of  friction  of  air.  He  found  that  the  rarefied 
air  in  a  supposed  vacuum  has  a  noteworthy  effect  on  a  pen- 
dulum. He  therefore  questioned  the  reliability  of  Baily's 
and  Sabine's  results.  Bessel's  work  he  considered  much 
better.  From  it  he  calculated  rj  for  air  =  .000275.  His  own 
observations  gave  y  =  .00027  to  .00052.  By  varying  the 
pressure  and  temperature  he  concluded  that  with  diminishing 
pressure  rj  decreases,  but  less  rapidly  than  the  density,  and 
with  rise  of  temperature  T)  increases  an  insignificant  amount, 
whereas  Maxwell  claimed  it  rises  2%  for  each  10°  C.  which  is 
not  so  far  from  the  truth. 

The  following  year  in  order  to  check  his  own  results,  O.  E. 
Meyer3  calculated  coefficients  of  friction  from  Graham's  ob- 


1.  Proc.  Royal  Society,  1865,  Vol.  14,  p.  339.      Phil.  Mag.  (4),  Vol.  30, 
p.  314. 

2.  Crelle's  Journal  fur  Math.,  1861,  Bd.  59,  p.  229.     Pogg.  Ann.,  1861, 
Bd.  113,  p.  55.     1865,  Bd.  125,  pp.  177,  401,  564. 

3.  Pogg.  Ann.,  1866,  Vol.  127,  pp.  253.  353. 


12 

servations,  although  he  believed  that  for  absolute  measure- 
ments Coulomb's  method  was  superior  to  Graham's,  because 
in  the  former  external  friction  is  eliminated,  it  being  settled 
that  the  gas  adheres  to  the  oscillating  disks.  For  the  trans- 
piration method  he  derived  the  formula  : 


in  which  /  is  the  time;  pl  is  the  pressure  of  the  gas  entering 
the  capillary;  p2  is  the  pressure  of  the  gas  leaving  the  cap- 
illary; V\  is  the  volume  of  the  transpired  gas  measured  at  the 
pressure  pv\  R  is  the  radius  of  the  capillary;  L  is  its  length; 
£  is  the  coefficient  of  slip  of  the  gas  on  the  walls  of  the  cap- 

£ 
illary.     The  term    4  ^   is  so  small  compared  with  1  that    most 

investigators  until  very  recently  have  neglected  it.  He  found 
T?  for  air  at  about  15°  C.  =  0.000178  to  0.000206  from  Graham's 
experiments.  He  was  fully  convinced  by  this  time  that  r?  is 
almost,  if  not  absolutely,  independent  of  the  pressure,  but  he 
was  not  certain  of  the  law  in  accordance  with  which  T?  increases 
with  the  temperature. 

In  1867  Lothar  Meyer1  published  his  calculations  of  the 
molecular  volumes  as  derived  from  his  brother's  determina- 
tions of  f.  He  derived  the  formula: 


in  which  v  =  molecular  volume;  C  =  a  then  not  obtainable 
constant  which  is  the  same  for  all  gases  at  the  same  tempera- 
ture; m  =  molecular  weight  of  the  gas.  He  showed  that  the 
ratio  of  the  molecular  volumes  of  two  substances  is  about 
the  same  as  the  ratio  of  their  molecular  volumes  as  determined 
by  Kopp2  from  their  boiling  points. 

Naumann,3  working  on  the  same  lines  that  year,  derived  the 

formula  — ^=    ^-O  — which  r  and  r1  are  in    the  radii  of  spheres 

T\  T)    \  mi 

of  action,  and  m  and  mi  are  the  molecular  weights  of  two  sub- 

1.  Ann.  d.  Chem.  u.  Phar.,  1867,  Suppl.  Bd.,  5,  p.  129. 

2.  Ann.  d.  Chem.  u.  Phar.,  1855,  Bd.  96,  pp.  1,  153,  303. 

3.  Ann.  d.  Chem.  u.  Phar.,  1867,  Suppl.  Bd.  5,  p.  252. 


13 

stances.     He  showed  the  methyl  ether  molecule  to  be  9  times 
the  volume  of  a  hydrogen  molecule. 

In  1871  O.  E.  Meyer1  returned  to  the  observations  of  pen- 
dulums and  made  some  careful  experiments  with  pendulums 
varying  in  length  from  14.5528  to  4.6868  metres,  which  were 
set  up  in  the  stairway  of  the  University  of  Breslau.  He  found 
i)  for  air  at  18°  C.  to  be  0.000216  to  0.000233. 

Recognizing  that  Maxwell's  modification  of  Coulomtys 
method  was  superior  to  his  own,  which  gave  only  the  \  r/ 
instead  of  T?,  and  was  less  sensitive  than  Maxwell's  owing  to 
the  absence  of  stationary  plates  between  the  oscillating  ones, 
Meyer'2  proceeded  now  to  repeat  Maxwell's  experiments.  In 
order  to  avoid  irregularities  in  the  internal  friction  of  a  single 
wire  he  adopted  bifilar  suspension.  He  found  T?  for  air  at 
18°  C.  =  0.000190  to  0.000197  which  he  considered  good 
agreement  with  Maxwell's  value  0.0002.  In  accordance  with 
these  more  correct  values  of  the  coefficient  of  viscosity  of  air 
he  revised  his  list  of  y  for  other  gases  and  gave  to  methyl  ether 
the  value  of  0.000107  at  15°  C. 

During  that  same  year  Lang3  published  the  results  of  some 
experiments  on  air  and  other  gases,  using  the  transpiration 
method.  His  apparatus  was  extremely  simple,  the  gas  being 
sucked  through  a  capillary  by  a  falling  column  of  water.  His 
capillaries  wrere  of  glass,  one  of  round  and  another  of  elliptical 
cross  section.  The  round  capillary  gave  ry  for  air  at  15°  0.000168 
to  0.000178;  the  elliptical  one  gave  at  9°  0.000143  to  0.000148. 
The  discrepancy  between  the  results  of  the  two  capillaries 
Meyer  suggested  might  be  partly  owing  to  the  fact  that  perhaps 
the  cross  section  of  the  elliptical  one  was  not  a  true  ellipse. 

Having  already  shown  theoretically  that  the  volume  of  gas 
delivered  by  a  capillary  in  a  unit  of  time  is  proportional 
to  the  fourth  power  of  its  radius,  inversely  proportional  to  its 
length,  and  directly  proportional  to  the  difference  in  pressure 
between  its  two  ends,  provided  the  gas  is  measured  at  a  pressure 
which  is  the  arithmetical  mean  betwreen  the  two  terminal 
pressures,  O.  E.  Meyer4  proceeded  in  1873  to  prove  experimen- 


1.  Pogg.  Ann.,  1871,  Vol.  142,  p.  481. 

2.  Pogg.  Ann.,  1871,  Vol.  143,  p.  14. 

3.  Wien.  Ber.  Mathem.  Naturw,  1871,  Vol.  63,  2  Abt.,  p.  604.  Wien. 
Ber.  Mathem.  Naturw,  1872,  Vol.  64,  2  Abt.,  p.  487.  Pogg.  Ann.,  1872, 
Vol.  145,  p.  290.  Pogg.  Ann.,  1873,  Vol.  148,  p.  550. 

4  Pogg.  Ann.,  1873,  Vol.  148,  pp.  1,  203. 


14 

tally  this  law,  whose  similarity  to  Poiseuille's  law  for  liquids 
is  at  once  evident.  He  employed  several  capillaries  varying 
in  cross  section  from  .0008  to  .0016  sq.  cm.  Working  under 
the  assumption  that  the  air  adheres  so  firmly  to  the  walls  of 
the  capillary  that  there  is  no  slipping,  he  obtained  the  fol- 
lowing values  of  y  for  air:  0.000168  at  0°  C.,  0.000184  at  14.4°  C., 
0.000197  at  21.1°C. 

The  above  experiments  were  made  by  causing  the  air  to  trans- 
pire from  one  copper  vessel  through  the  capillary  into  another 
similar  vessel,  allowing  the  pressure  but  not  the  volume  of  the 
air  to  change.  He  next  repeated  the  experiments  by  allowing 
the  volume  but  not  the  pressure  to  change  by  employing  con- 
stant water  suction.  He  found  T?  at  10°  C.  to  be  .000192- 
.000199  and  at  100°  C.  r?  =  .000211  -  .000218.  His  results  do 
not  agree  with  Maxwell's  in  showing  T?  to  vary  as  the  absolute 
temperature,  neither  do  the}7  agree  with  theory  in  proving  T) 
to  vary  as  the  square  root  of  the  absolute  temperature.  His 
results  are  midway  between  Maxwell's  and  the  theory,  viz., 
r)  varies  as  the  3/4  power  of  the  absolute  temperature.  He 
proposed  the  formula: 

7?  -  >?0(1+ 0.0024  T) 

in  which  T?O  =  0.000171  and  T  =  temperature  in  C°.  Maxwell's 
formula  was  i?  =  T?O  (1+0.003665  T). 

Using  a  still  smaller  capillary  whose  cross  section  was  .00015 
sq.  cm.  through  which  he  sucked  the  air  by  means  of  a  mercury 
pump  he  found  >?  at  22°  C.  to  be  .000187  and  at  100°  C.  .000223 
.000227.     He  then  modified  his  formula  to  read  y  =  0.000174 
(1+0.003  T). 

Repeating  Maxwell's  oscillation  experiments  once  more  with 
bifilar  suspension  Meyer  found  y  =  .000190  (1+0.0025  T)  which 
is  in  fairly  good  agreement  with  his  transpiration  results,  so 
that  Meyer  concludes  Maxwell's  results  are  in  error  owing  to 
the  heating  of  the  suspension  wire  and  the  poor  position  of 
the  thermometer  with  reference  to  the  oscillating  disks.  Meyer 
accounts  for  the  difference  between  his  own  results  and  theory 
by  the  fact  that  heat  increases  not  only  the  rectilinear  velocity 
of  the  molecules  but  also  the  internal  movements  of  the  atoms 
which  build  up  the  molecules. 

A  little  later  Meyer  and  Springmuhl1  using  the  same  apparatus 
obtained  the  coefficients  of  friction  of  19  different  gases,  some 


1.   Pogg.  Ann.,  1873,  Vol.  148,  p.  526. 


15 

of  which  had  been  investigated  by  Graham,  with  whom  they 
agreed  within  2  to  20%.  Their  values  for  air  and  methyl 
ether  were  .000190  and  .000102  at  15°  C. 

No  difference  in  y  was  observed  when  transpiration  took 
place  into  a  vessel  filled  with  the  same  kind  of  gas  or  with  a 
different  gas,  showing  that  there  was  no  diffusion  back  through 
the  capillary.  This  proves  that  a  gas  has  the  same  pressure 
upon  another  kind  of  gas  as  upon  the  same  kind  as  itself. 

The  following  year  Puluj1  investigated  the  coefficient  of  in- 
ternal friction  of  air  as  a  function  of  the  temperature,  using 
a  transpiration  apparatus  like  Professor  Lang's.  He  main- 
tains that  the  loss  of  heat  due  to  the  expansion  of  the  air  in 
the  capillary  is  compensated  for  by  the  heat  which  the  air 
receives  from  the  walls  of  the  capillary.  He  found  y  for  air 
at  15°  C.  =  0.00018526.  For  a  temperature  range  from  13.4°  C. 
to  27.2°  he  found  y  to  vary  according  to  the  formula  y  =  r?0 
(1+0.003665  r)°-G52776±  0-020893. 

After  enlarging  the  temperature  range  from  1.5°  C.  to 
91.2  he  found  the  exponent  of  the  above  parenthesis  changed  to 
0.590609  ±0.009510. 

Kundt  and  Warburg2  published  in  1875  an  important  in- 
vestigation which  proves  that  the  law  that  y  is  independent 
of  the  pressure  does  not  hold  for  extremely  small  pressures. 
They  used  Coulomb's  method,  employing  only  one  oscillating 
glass  disk  with  bifilar  suspension,  For  air  at  15°  C.  they 
found  y  =  0.000189  and  for  pure  aqueous  vapor  at  15°  C.  and 
a  tension  of  about  16  mm.  y  =  .000099. 

They  found  that  with  air  the  logarithmic  decrement  began 
to  diminish  at  a  pressure  of  2|  mm.  of  mercury,  and  fell  off 
very  perceptibly  at  a  pressure  of  ^  mm.  This  diminution 
was  ascribed  to  a  slipping  of  the  air  on  the  surface  of  the  oscil- 
lating disk.  From  their  work  they  concluded  that  the  coefficient 
of  slip  for  a  gas  on  a  solid  partition  has  sensibly  a  determined 
value  dependent  upon  the  nature  of  the  gas,  so  long  as  the 
latter  is  present  in  layers  thicker  than  14  times  the  mean  length 
of  molecular  path,  and  it  is  inversely  proportional  to  the  pres- 
sure. The  absolute  value,  which  is  the  coefficient  of  internal 


1.  Wien.    Ber.    Mathm.    Naturw.,    1874,    Vol.   69    (2),    p.    287.     Wien. 
Ber.  Mathm.  Naturw.,  1874,  Vol.  70  (2),  p.  243. 

2.  Monatsber.  d.  Berl.  Akad.,   1875,  p.   160.     Pogg.  Ann.,   1875,  Vol. 
150,  pp.  337,  525.     Phil.  Mag.,  1875  (4),  Vol.  50,  p.  53. 


16 

friction   divided   by   the   coefficient   of  external   friction   of  the 

760 
gas,  they  found  from  experiment  to  te  for  air  0.0001-—    whereas 

760 
theory  showed  it  ought  to  be  0.000058 where    p  =   pressure 

expressed  in  mm.  of  mercury. 

Warburg1,  pursuing  the  same  subject  further  the  following 
year,  proceeded  to  find  the  coefficient  of  slip  for  air  by  the 
transpiration  method.  He  pointed  out  that  because  the  radius 
of  a  capillary  is  smaller  than  we  can  make  the  distance  between 
oscillating  plates,  the  effect  of  slipping  in  transpiration  experi- 
ments is  noticeable  at  higher  pressures  than  in  the  method 
of  oscillations.  Using  a  pressure  of  38  mm.  of  mercury  and  a 
capillary  of  0.15  mm.  radius  he  found  r?  for  air  was  5%  smaller 
than  it  ought  to  be,  owing  to  slipping.  The  temperature 
exponent  from  0°  to  100°  C.  he  found  to  be  0.77  which  is  in 
good  agreement  with  O.  E.  Meyer.  His  temperature  exponent 
for  hydrogen  is  0.63.  He  gives  the  formula: 

-   =  1  H — 5-  where    £  =    coefficient    of    slip     and  R 
Observed  y  R 

=  radius  of  the  capillary. 

Obermayer's2  results  agree  with  those  of  O.  E.  Meyer  in  showing 
that  f)  for  air  increases  as  the  3/4  power  of  the  absolute  tem- 
perature. He  used  the  transpiration  method  and  pointed  out 
two  possible  sources  of  error  inherent  in  this  method:  1st.  the 
clinging  of  a  layer  of  air  to  the  inside  wall  of  the  capillary 
which  diminishes  the  cross  section  more  at  low  than  at  high 
temperatures;  2nd.  the  change  in  temperature  which  the  gas 
undergoes  in  the  capillary  as  it  expands  from  the  entering  to 
the  terminal  pressure. 

He  carried  the  temperature  as  high  as  270°  C.  and  criticized 
Puluj  for  his  small  temperature  range.  His  absolute  values 
of  T)  are  lower  than  those  of  Meyer.  He  points  out  that  the 
so-called  permanent  gases  are  distinguished  by  a  temperature 


1.  Pogg.  Ann.,  1876,  Vol.   159,  p.  379. 

2.  Wien.   Ber.   Mathem.    Naturw.,    1875,   Vol.   71    (2),   p.   281.     Wien. 
Her.  Mathem.  Naturw.,  1876,  Vol.  73  (2),  p.  433.     Carls  Rep.,  1876  (2), 
Vol.  12,  pp.  13,  456.     Carls  Rep.,  1877,  Vol.  13,  p.  130.     Phil.  Mag.,   1886, 
Vol.  21. 


17 

exponent  of  3/4,  whereas  the  coercible  gases  have  this  exponent 
almost  unity. 

E.  Wiedemann1,  assuming  as  did  several  predecessors  that  the 
variation  of  T)  with  the  temperature  can  be  expressed  by  the 
formula  TJ  =  T?O  (1-f  a  T)n,  proceeded  to  determine  the  exponent 
n,  which  I  call  the  temperature  exponent,  for  several  gases. 
He  adopted  the  transpiration  method,  driving  the  gas  through 
the  capillary  by  the  pressure  produced  by  mercury  flowing 
from  a  reservoir.  His  capillary  was  surrounded  by  cold  water, 
steam  or  aniline  vapor,  so  that  his  temperature  range  was 
from  0  to  184.5°  C.  He  assumed  that  the  absolute  value  of  7) 
for  air  had  already  been  determined  with  sufficient  accuracy 
by  Meyer,  Maxwell,  Kundt,  Warburg  and  others,  so  that  he 
gave  only  comparative  results.  He  found  the  temperature 
exponent,  n,  to  vary  with  different  gases  but  in  different  ways. 
With  most  gases  he  tried,  it  decreased  with  higher  tempera- 
tures. Thus  for  air  from  0°  to  100°  C.  it  was  .7333  and  from 
100  to  184.5  it  was  only  .6701.  For  CO  it  was  .6949  from 
0°  to  184.5°  C. 

Puluj2  with  a  modified  Coulomb  apparatus  consisting  of  an 
oscillating  disk  of  thin  mirror  glass  between  two  fixed  disks 
of  thick  plate  glass,  found  the  temperature  exponent  for  air 
=  0.72196  ±0.01825  which  agrees  pretty  well  with  Meyer  and 
Obermayer  but  not  with  his  own  previous  results. 

Kirchhoff3  in  1877  gave  a  complete  mathematical  deduction 
of  the  formulae  to  be  used  in  calculating  the  coefficient  of  fric- 
tion of  a  gas  from  the  logarithmic  decrement  of  a  sphere  oscil- 
lating about  a  diameter  and  for  an  ellipsoid  of  revolution  ro- 
tating about  an  axis  of  symmetry. 

Holman,4  in  a  series  of  very  careful  experiments  by  the 
transpiration  method,  found  that  y  for  air  increases  as  the  0.77 
power  of  the  absolute  temperature.  Later  experiments  at 
higher  temperatures  showed  that  the  increase  of  T?  falls  off 
in  rate  as  the  temperature  rises,  except  possibly  in  the  case 
of  hydrogen,  which  he  therefore  regards  as  oar  most  perfect  gas. 

Guthrie5    in    order   to    determine    whether    there    were    any 


1.  Archiv.   d.   Sc.    Phys.   et    Xat.   de  Geneve,    1870,    Vol.    56,    p.     277 
Fortschr.  d.   Phys.,   1876,  Vol.  32,  p.  200. 

2.^Wien.  Ber.  Mathem.  Naturw.,  1876,  Vol.  73  (2).  p.  589. 
:.{.  \\Iechanik,  1877,  4  Aufl.,  26  Vorl.,  p.  383. 

4.  Proc.  Am.  Acad.,  Boston,  1877,  Vol.  12,  p.  41;  1886,  Vol.  21.  p.  1. 
Phil.  Mag.,  1877  (5),  Vol.  3,  p.  81;  1886,  Vol.  21,  p.  199. 

5.  Phil.  Mag.,  1878  (5),  Vol.  5,  p.  433. 


disturbances,  such  as  eddies,  in  the  flow  of  a  gas  at  the  begin- 
ning and  end  of  a  capillary,  took  the  transpiration  time  of  a 
given  volume  of  air  through  a  capillary,  that  he  afterward  cut 
up  into  a  number  of  small  pieces  which  he  joined  together 
by  guttapercha  tubing.  With  the  capillary  in  this  condition 
he  found  the  same  transpiration  time  as  before,  which  seemed 
to  prove  the  absence  of  disturbances  at  the  ends  of  the  cap- 
illary. A  later  investigator,  Hoffmann,  does  not  agree  with  him. 

Puluj1  in  1878  attacked  the  problem  of  the  internal  friction 
of  vapors,  using  the  modified  Coulomb-Maxwell  apparatus 
already  mentioned,  with  unifilar  suspension.  From  experi- 
ments with  ether  he  concluded  that  its  TJ  is  independent  of  the 
pressure.  He  found  T?  at  10°  =  0.0000716  and  at  37.1°  C." 
=  0.0000792.  Its  variation  with  the  temperature  he  gives  by 
the  equation 

7?T  =  0.0000689  (1  +  0.0041575  T)  °'94 

The  coefficient  of  expansion  of  ether  =  .0041575  he  calculated 
from  Herwig's2  data.  From  this  equation  he  concluded  that 
for  ether  vapor  and  probably  all  vapors  ^  is  proportional  to  the 
absolute  temperature.  He  also  experimented  on  the  vapors 
of  alcohol,  chloroform,  benzol  and  aceton.  Taking  the  mole- 
cular volume  of  hydrogen  as  1,  he  found  those  for  ether  and 
alcohol  101.1  and  52.9  respectively,  which  is  in  fair  agreement 
with  Kopp's3  molecular  volumes  deduced  from  the  boiling 
points  of  the  liquids. 

Puluj  agreed  with  Stefan4  in  thinking  that  each  molecule 
of  a  gas  is  surrounded  by  an  envelope  of  ether,  and  in  explain- 
ing the  increase  of  internal  friction  with  rise  of  temperature, 
by  saying  that  with  increased  velocity  the  molecules  on  impact 
penetrate  further  into  one  another.  His  experiments  with  air 
at  very  low  pressures  led  to  the  result,  that  while  the  pressure 
diminishes  from  754  to  0.03  mm.,  the  coefficient  of  friction  only 
decreases  to  about  one  half  its  value,  which  proves  what  a 
great  number  of  molecules  remain  in  quite  high  vacuum. 

Puluj5  next  took  up  the  problem  of  determining  the  co- 
efficient of  friction  of  a  mixture  of  the  two  gases  CO2  and  H,  using 


1.  Wien.  Ber.,  1878,  Vol.  78  (2),  p.  279;  Carls  Rep.,  1878,  Vol.  14,  p. 
573;  Phil.  Mag.,  1878  (5),  Vol.  6,  p.  157. 

2.  Pogg.  Ann.,  1869,  Bd.  137,  p.  595. 

3.  Ann.  Chem.  Phar.,  1855,  Bd.  96,  pp.  1,    153,     303. 

4.  Wien.  Ber.,  1862  (2),  Vol.  46,  pp.  8,  495;  1872  (2),  Vol.  65,  p.  360. 

5.  Carls  Rep.,  1879,  Vol.  15,  p.  578  and  633. 


19 

the  same  apparatus  as  for  vapors.  He  drew  the  conclusions: 
(1)  The  coefficient  of  friction  of  a  mixture  of  CO2  and  H  is  not 
larger  (smaller)  than  the  coefficient  of  that  constituent  which 
has  the  larger  (smaller)  coefficient;  (2)  gases  with  larger  mole- 
cular weights  have  in  a  mixture  of  equal  proportions  a  greater 
influence  on  the  value  of  the  coefficient  of  friction  of  the  mixture. 

He  proved  that  Maxwell's  formula  for  the  y  of  a  mixture 
gives  too  small  a  value,  and  proposed  another  formula  which 
gives  ry  too  large,  but  agrees  with  experiment  in  showing  the 
curious  fact  that  with  a  small  percentage  of  the  lighter  gas  >? 
increases. 

In  1879  Lothar  Meyer1  began  the  study  of  the  internal  fric- 
tion of  vapors  by  the  transpiration  method.  The  capillary, 
which  was  nearly  1£  metres  in  length,  was  coiled  into  a  helix 
and  fastened  in  the  upper  part  of  a  boiling  flask.  The  lower 
end  of  the  capillary  passed  through  the  neck  of  the  flask  into 
a  condenser.  The  substance  to  be  examined  was  boiled  at  a 
regulated  pressure,  the  vapor  evolved  surrounding  the  cap- 
illary and  raising  it  to  the  same  temperature.  A  part  of  the 
vapor,  which  was  of  course  saturated,  entered  the  upper  end 
of  the  capillary  and  passed  through  it  into  an  air-free  cooled 
space,  where  it  was  condensed  and  measured  as  a  liquid.  The 
volume  of  the  vapor  transpired  was  calculated  from  the  amount 
of  liquid  collected  in  the  condenser. 

This  method  is  open  to  criticism  for  several  reasons,  one 
of  which  is  the  coiling  of  the  capillary  into  a  small  helix,  thus 
possibly  deforming  its  bore  and  causing  additional  resistance 
by  the  curved  path  of  the  gas.  The  vapor  at  the  boiling  point 
of  the  liquid  can  also  hardly  be  regarded  as  a  true  gas,  and 
should  have  been  tested  at  several  degrees  above  boiling.  The 
expansion  of  the  saturated  vapor  as  it  passed  through  the 
capillary  probably  also  changed  its  state.  Furthermore  the 
great  length  of  time,  several  hours,  needed  for  transpiration 
made  it  difficult  to  keep  conditions  constant. 

A  preliminary  test  with  air  gave  y  at  room  temperature 
=  0.000188  which  is  higher  than  most  other  observers  have 
found  by  the  transpiration  method.  He  found  y  for  benzole 
about  50%  higher  than  Puluj  had  found,  that  its  increase  with 
rise  of  temperature  is  more  rapid  than  for  so-called  permanent 
gases.  He  discovered  that  with  a  difference  in  pressure  of 


1.   Wien.  Ann.,   1879,  Bd.,  7  p.  497. 


20 

14  cm.  of  mercury  between  the  two  ends  of  the  capillary  his 
apparatus  gave  y  considerably  too  small.  On  calculating 
molecular  volumes  he  found  that  they  are  larger,  the  lower 
the  temperature. 

Two  years  later  were  published  the  results  of  observations 
made  by  L.  Meyer  and  Schumann1  on  a  very  large  number 
of  substances,  using  the  transpiration  apparatus  just  described. 
Two  capillaries  were  employed,  one  1427  mm.  long  and  0.31  mm. 
in  diameter;  the  other  1404  mm.  long  and  0.3328  mm.  in  diam- 
eter. The  second  capillary  gave  values  3%  higher  than  the 
first.  Below  is  the  table  of  their  results  for  Esters. 

TABLE  V.     ESTERS  Cn  H2n  O2    7)  X   10  6. 
Acid  Radical  Alcohol  Radical 


Methyl 

Ethyl 

Propyl 

Isobutyl 

Amyl 

n  =  2 

3 

4 

5 

6 

Formic  Acid 

173 

156 

159 

172 

160 

n  =  3 

4 

5 

6 

7 

Acetic  Acid 

152 

152 

160 

155 

n  =  4 

5 

6 

7 

8 

Propionic  Acid 

150 

158 

153 

164 

158 

n  =  5 

6 

7 

8 

9 

Normal  Butyric  Acid 

159 

160 

164 

167 

155 

Iso-Butyric  Acid 

152 

151 

153 

158 

155 

w  =  6 

7 

8 

9 

10 

Valerianic  Acid 

163 

165 

167 

154 

They  conclude  that  all  esters  at  their  boiling  points  and 
at  the  same  pressure  transpire  very  nearly  equal  volumes  of 
vapor,  which  however  because  the  boiling  points  are  different, 
do  not  contain  the  same  number  of  molecules. 

They  found  rj  for  the  corresponding  acids  considerably  smaller 
than  for  their  esters.  They  believed  that  the  differences  of  y 
are  partly  too  small  and  partly  too  irregular  to  draw  any  sure 
conclusion  of  the  dependence  of  r?  on  the  molecular  constitu- 
tion. They  point  out,  however,  some  differences  which  seem 
quite  regular. 

The  esters  of  acetic,  propionic  and  isobutyric  acids  show 
almost  always  a  smaller  r?  than  those  of  formic,  normal  butyric 


1.   Wied.  Ann.,  1881,  Bd.  13,  p.  1. 


21 

and  valerianic  acids.  This  difference  is  especially  noticeable 
in  the  two  butyric  acids.  The  influence  of  the  alcohol  radical 
cannot  be  seen  so  clearly.  Among  the  isomeric  esters,  those 
of  formic,  acetic,  propionic  and  isobutyric  acids  have  the  greater 
TI  with  the  greater  alcohol  radical.  The  isomeric  esters  of 
normal  butyric  and  valerianic  acids  do  not  follow  this  rule  but 
have  the  same  friction. 

The  number  of  carbon  atoms  in  a  molecule  also  seems  to  have 
a  certain  influence  on  i).  The  esters  for  which  n  =  2,  5,  7  or  8 
show  a  larger  T?  than  those  with  3,  4  or  9  carbon  atoms.  The 
cause  of  these  differences  they  were  unable  to  explain. 

On  calculating  the  relative  molecular  volumes  they  found 
them  only  about  half  those  given  by  Kopp's  rule.  L.  Meyer 
tried  to  account  for  this  by  advancing  the  hypothesis  that 
by  Kopp's  method  the  empty  space  is  included  which  is  open 
to  the  atoms  for  their  motion,  while  from  the  coefficient  of 
internal  friction  only  the  volume  of  the  gas  particles  themselves 
is  determined. 

Steudel1  continued  the  work  of  L.  Meyer  and  Schumann, 
using  the  same  apparatus  with  the  second  capillary,  the  first 
having  been  broken.  He  investigated  several  homologous 
lines  of  organic  compounds,  viz.,  alcohols  up  to  four  atoms 
of  carbon  per  molecule,  and  their  halogen  derivatives,  also  some 
substitution  products  of  ethane  and  methane. 

He  found  the  transpiration  time  increased  with  the  mole- 
cular weight.  Of  isomeric  compounds  at  the  boiling  points,  the 
normal,  i.e.,  those  that  boil  at  the  highest  temperature  trans- 
pire the  slowest  and  the  tertiary  the  fastest,  with  the  exception 
of  isopropyl  alcohol  which  transpires  noticeably  slower  than 
the  normal.  Unsymmetrical  low  boiling  compounds  have  a 
smaller  transpiration  time  than  the  symmetrical.  The  only 
exception  he  found  to  the  rule  that  the  transpiration  time  in- 
creases with  the  molecular  weight  is  methyliodide  whose  time 
was  1037  minutes,  which  is  almost  the  same  as  1056  minutes 
taken  by  isobutyliodide. 

He  points  out  that  the  coefficients  of  friction  of  each  line 
of  homologous  compounds  are  nearly  alike  or  only  slightly 
different.  The  values  for  the  primary  alcohols  vary  from 
.000135  to  .000143;  for  three  of  them  they  are  almost  exactly 
alike.  The  isopropyl  gave  a  considerably  larger  figure,  also 


1.   Wied.  Ann.    1882,  Bd.   16,  p.  369. 


22 
TABLE  OF  rj  X  106. 


Radical 

Alcohol 

Chloride 

Bromide 

Iodide 

Methyl 

135 

116* 

245 

Ethyl 

142 

105* 

183 

216 

Normal  Propyl 

142 

146 

184 

210 

Isopropyl 

162 

148 

176 

201 

Normal  Butyl 

143 

149 

202 

Isobutyl 

144 

150 

179 

204 

Tertiary  Butyl 

160 

150 

*  Calculated  from  Graham's  results. 

the  tertiary  butyl.  The  chlorides  seem  to  all  have  the  same 
friction.  The  greatest  differences  exist  in  the  iodide  column. 

He  found  a  considerable  difference  in  TJ  with  a  change  in 
pressure.  With  a  greater  pressure  than  10  to  15  cm.  of  mercury 
the  formula  seemed  to  lose  its  validity  for  vapors.  This  agrees 
with  what  L.  Meyer  had  noticed  with  benzole. 

Steudel  also  gives  tables  of  molecular  velocities,  mean  free 
paths,  combined  cross  section  of  molecules  and  cross-sections  of 
an  equal  number  of  molecules.  He  points  out  that  the  cross 
sections  of  molecules  of  isomeric  compounds  are  not  the  same. 
For  the  butyl  compounds  the  areas  for  the  normal  substances 
are  the  largest,  the  tertiary  the  smallest,  with  the  iso  com- 
pounds between.  Propyl  and  isopropyl  alcohol,  as  also  both 
chlorides  show  the  same  relation.  On  the  other  hand,  isopropyl- 
bromide  has  a  greater  molecular  area  than  the  normal  com- 
pound. In  the  substitution  products  of  ethane  the  sym- 
metrical ones  have  a  larger  area  than  the  unsymmetrical. 

Steudel  also  calculated  the  relative  molecular  volumes  from 
SO2  according  to  the  method  of  L.  Meyer,  with  whom  he  agrees 
in  finding  the  volumes  about  one  half  as  large  as  those  cal- 
culated according  to  Kopp's  rule.  The  ratios  of  molecular 
volumes  of  the  different  substances,  however,  are  almost  the 
same  as  the  ratios  of  the  volumes  according  to  Kopp. 

L.  Meyer,1  summing  up  Steudel's  and  his  own  previous 
results,  gives  the  following: 

TABLE  OF  )?X106 

Alcohols  CnH2n+2O 142  average 

Chlorides  CnH2n+1Cl 150 

Esters  CnH2nO2 155 

Bromides  C^H^+jBr 182 

Iodides  CnH2n+11 210        "j| 


1.   Pog£.  Ann.,  1882,  Vol.  16,  p.  394. 


23 

Cases  in  which  n  =  1  and  some  few  unaccountable  varia- 
tions are  omitted  from  the  above.  When  n  =  1  the  variations 
are  great  but  he  could  see  no  law  of  the  influence  of  molecular 
constitution  on  the  friction. 

On  the  other  hand  the  influence  of  the  nature  of  the  atom 
is  very  clearly  seen;  friction  of  iodine  > bromine > chlorine. 

The  molecular  volumes  do  not  agree  with  those  L.  Meyer1 
calculated  from  Graham's  results.  He  points  out  that  this 
one  fact  seems  certain,  viz. :  the  molecules  of  a  tertiary  butyl 
compound  are  smaller  than  those  of  a  secondary  which  are 
smaller  than  those  of  a  primary.  This  is  in  agreement  with 
the  universal  view  taken  of  the  concatenation  of  these  com- 
pounds. Those  of  the  tertiary  are  grouped  around  a  single 
atom  of  carbon,  hence  are  more  spherical  in  shape.  Propyl 
and  isopropyl  compounds  on  the  other  hand  do  not  show  the 
same  regularity.  The  alcohols  and  iodides  deviate  in  opposite 
directions,  which  is  unexplained.  He  further  points  out  that 
the  sphere  of  action  of  a  liquid  molecule  increases  with  the 
temperature,  while  that  of  a  gaseous  molecule  diminishes  with 
rise  of  temperature. 

In  1881  Crookes2  published  an  interesting  account  of  ex- 
periments on  the  viscosity  of  gases  at  high  exhaustions.  His 
method  consisted  in  observing  the  logarithmic  decrement  of  a 
plate  of  mica  enclosed  in  a  glass  bulb,  the  axis  of  suspension 
being  in  the  plane  of  the  mica.  He  found  that  with  air,  nitro- 
gen, oxygen  and  carbon  monoxide  y  diminishes  slightly  as  the 
pressure  falls  from  760  to  3  mm.,  after  which  it  decreases  very 
rapidly.  Hydrogen  showed  no  change  in  r)  from  760  to  3  mm. 
pressure.  When  the  pressure  was  \  mm.  the  mean  free  path 
of  a  molecule  became  comparable  with  the  dimensions  of  the 
glass  bulb,  and  the  ultra  gaseous  state  of  matter,  as  Crookes 
names  it,  was  assumed. 

Margules3  suggested  a  new  experimental  method  of  finding 
the  internal  friction  of  a  fluid  or  gas.  He  proposed  two  coaxial 
cylinders,  one  rotating  at  a  constant  speed,  the  other  having 
its  turning  moment  measured  directly.  He  also  pointed  out 
that  in  Coulomb's  method  the  molecules  near  the  rotating 
plate  do  not  move  in  circles  but  in  zigzags,  the  centrifugal 


1.  Liebig's  Ann.,  1867,  Suppl.  Bd.  5,  p.   129. 

2.  Phil.  Trans.,  1881,  Vol.  172,  p.  387. 

3.  Wien.  Ber.  Mathem.  Naturw.,  1881,  Bd.  83  (2),  p.  588. 


24 

force  carrying  them  outward  from  the  axis.  This  action  Maxwell 
had  apparently  neglected. 

Instead  of  an  oscillating  system  of  plates  Braun  and  Kurz1 
employed  a  sphere,  the  mathematical  theory  of  which  had 
been  worked  out  by  Kirchhoff.  They  found  y  for  air  at  room 
temperature  =  0.000184  ±  .000010. 

Warburg  and  Babo2  investigated  the  relation  between  vis- 
cosity and  density  of  fluid,  especially  gaseous  substances. 
They  employed  the  capillary  method,  and  experimented  on 
liquid  CO2  and  gaseous  CO2  above  the  critical  temperature, 
and  at  pressures  between  30  and  120  atmospheres.  They 
found  that  the  viscosity  of  liquid  CO2  increases  with  the  den- 
sity, as  does  also  that  of  gaseous  CO2  at  very  high  pressures. 
The  fact  that  investigators  have  shown  that  the  viscosity  of  a 
gas  is  not  independent  of  the  pressure,  when  the  latter  is  either 
very  low  or  very  high,  does  not  destroy  the  validity  of  the 
law  for  ordinary  pressures. 

Grossman,3  noticing  that  the  transpiration  and  oscillation 
methods  are  not  in  perfect  agreement  in  their  values  of  >?, 
undertook  to  discover  the  cause  of  the  difference.  He  chose 
Coulomb's  method  for  his  work  as  being  the  simplest.  He 
proposes  corrections  within  certain  limits  and  deduces  a  for- 
mula which  he  thinks  applies  accurately  within  those  limits. 

Klemencic4  in  1881  published  a  complete  mathematical 
treatment  of  the  damping  of  the  oscillations  of  a  solid  body 
in  liquids  and  gases.  He  discusses  the  cases  of  spheres  and 
cylinders  oscillating  and  swinging  in  various  manners. 

In  order  to  help  settle  the  question  of  why  the  coefficient 
of  friction  of  a  gas  increases  with  the  temperature  Koch5  under- 
took the  study  of  the  coefficient  of  friction  of  mercury  vapor 
and  its  dependence  on  the  temperature.  Stefan's  hypothesis 
of  an  ether  envelope  of  the  molecule  has  already  been  men- 
tioned. O.  E.  Meyer's  explanation  that  at  higher  temperatures 
the  bonds  between  the  atoms  are  loosened  so  that  the  mole- 
cules on  collision  penetrate  further  into  each  other,  thus  dimin- 


1.  Carl  Rep.,  1882,  Bd.  18,  pp.  569,  665,  697;  1883,  Bd.  19,  pp.  343,  605. 

2.  Wied.  Ann.,  1882,  Bd.  17,  p.  390. 
.",.   Wied.  Ann.,   1882,  Bd.   16,  p.  619. 

4.  Wien.  Ber.  Mathem.  Naturw.,  1881,  Bd.  84  (2),  p.  146. 
Carl's  Rep.,  1881,  Bd.  17,  p.  144. 

Ann.  d.  Phys.  Beiblatter,  1882,  Bd.  6,  p.  66. 

5.  Wied.  Ann.,  1883,  Bd.  19,  p.  857. 


25 

ishing  the  distance  between  their  centres  and  thus  their  diam- 
eters, does  not  hold  in  the  case  of  a  monatomic  gas  such  as  mer- 
cury. Koch  employed  two  capillaries,  one  having  a  radius 

=  0.004245  cm.  and  length  =  9.875  cm.;  and  the  other  a 
radius  =  0.00593  cm.  and  length  =  19.22  cm. 

He  found  TJ  =  0.000494  at  0°  C.  and  y  =  0.000643  at  98°  C. 
With  a  third  capillary  whose  radius  =  .008238  cm.  and  length 

=  18  cm.  he  found  y  4%  larger  than  the  above,  hence  he  con- 
cludes Poiseuille's  law  does  not  hold  for  a  capillary  of  this  size 
and  neglects  this  result. 

Assuming  the  usual  equation  7?  =  JJQ  (\  +  at)n  he  found  the 
exponent  n  to  be  1.6  for  mercury  from  which  it  appears  that 
the  diameter  of  a  mercury  molecule  is  about  proportional  to 

j^.  where  T  =  absolute  temperature.     At  0°  C.  he  found  the 

molecular  volume  of  mercury  to  be  12.9  times,  and  at  300° 
only  4.39  times  that  of  hydrogen. 

In  1884  O.  Reynolds1  read  an  interesting  and  suggestive 
paper  on  the  two  manners  of  motion  of  water,  the  steady  or 
direct,  and  the  sinuous  or  eddying.  He  experimented  on 
water  flowing  through  glass  tubes  and  made  the  character  of 
motion  visible  by  means  of  color  bands  in  the  water.  In 
order  that  motions  in  tubes  of  different  sizes  can  be  compared, 
the  velocities  must  be  inversely  as  the  tube  diameters.  The 
critical  velocity  at  which  sinuous  motion  begins  increases 
with  the  viscosity  of  the  fluid.  If  water  be  flowing  in  a  bent 
channel  in  steady  streams,  the  question  as  to  whether  it  will 
remain  steady  or  not  turns  on  the  variation  in  the  velocity 
from  the  inside  to  the  outside  of  the  stream.  He  enumerates 
viscosity,  converging  solid  boundaries,  curvature  with  the 
velocity  greatest  on  the  outside  as  conducive  to  direct  or  steady 
motion,  whereas  diverging  solid  boundaries  and  curvature 
with  the  velocity  greatest  on  the  inside,  tend  to  sinuous  or 
unsteady  flow.  It  is  possible  that  similar  actions  take  place 
in  the  flow  of  a  gas  through  a  capillary,  and  constitute  an 
objection  to  a  coiled  capillary. 

Hoffmann2  worked  with  gases  along  somewhat  the  same 
lines  that  Reynolds  did  with  liquids.  Poiseuille's  law  takes 


1.  Proc.  Roy.  Inst.  Grt.  Brit.,   1884,  Vol.   11,  p.  44. 

2.  Wied.  Ann.,  1884,  Bd.  21,  p.  470. 


26 

for  granted  that  the  particles  of  gas  move  through  a  capillary 
in  parallel  lines.  Hoffmann  tried  to  prove  that  when  this 
law  does  not  apply  it  is  because  the  molecules  have  a  sinuous 
motion  through  the  capillary.  He  thought  that  the  disturbing 
cause  resides  chiefly  at  the  beginning  and  end  of  the  cap- 
illary, although  Guthrie's  results  seemed  to  contradict  this. 
Hoffmann  repeated  Guthrie's  experiments  and  found  a  longer 
transpiration  time  for  the  many  small  pieces  than  when  the 
capillary  was  all  in  one  piece.  He  points  out  that  perhaps 
Guthrie  worked  with  too  low  pressures  and  put  the  pieces  of 
capillary  too  close  together. 

In  order  to  study  the  whirling  motion  he  drove  tobacco 
smoke  through  tubes  (of  course  not  capillary)  and  found  a 
conical  contraction  of  flow  at  the  end,  and  further  on  a  conical 
spreading  out  in  which  could  be  seen  powerful  eddies.  The 
eddies  approached  nearer  to  the  end  as  the  velocity  was  in- 
creased, until  finally  the  contraction  vanished  and  the  eddies 
were  right  at  the  end  of  the  tube.  These  observations  agree 
with  those  of  Sondhaus.1  As  a  result  of  all  his  work  Hoffmann 
concludes  that  when  Poiseuille's  law  does  not  hold  for  a  tube, 
the  chief  reason  is  the  phenomena  at  the  end  and  especially 
at  the  beginning  of  the  capillary. 

As  it  was  impossible  to  determine  by  the  method  employed 
in  1881  by  L.  Meyer  and  Schumann,  the  dependence  of  the 
friction  of  vapors  on  the  temperature,  Schumann2  proceeded 
to  make  this  investigation  by  the  oscillation  method.  His 
apparatus  was  similar  to  Maxwell's,  but  he  found  that  Maxwell's 
formula  did  not  give  concordant  results,  so  he  adopted  an 
empirical  formula  which  gave  values  in  good  agreement,  but 
generally  smaller  than  those  of  other  observers  excepting 
Obermayer.  His  y  for  air  at  20°  C.  =  .000178  and  at  0°  C. 
.000168.  His  results  are  in  fair  accord  with  those  of  the  trans- 
piration method  at  ordinary  temperatures.  At  high  tempera- 
tures he  claims  that  capillaries  give  values  which  are  too  small 
due  to  the  adsorption  of  the  gas  by  the  capillary  walls. 

To  express  the  increase  of  y  with  rise  of  temperature  he  pro- 
poses a  new  formula: 


1.  Pogg.  Ann.,  1852,  Bd.  85,  pp.  58. 

2.  Ann.  d.  Phys.,  1884,  Bd.  23,  p.  353. 


27 

in  which  a  =  coefficient  of  expansion  of  the  gas,  and  y  =  co- 
efficient of  diminution  of  the  radius  of  the  sphere  of  action  of 
the  molecule.  For  air  a  =  0.003665,  y  =  0.000802.  For 
CO,  a  =0.003701,  y  =0.000889.  For  all  vapors  a  is  .004, 
but  for  benzole  y  =  0.00185  while  for  other  vapors  it  is  0.00164. 

He  found  rj  entirely  independent  of  the  pressure  except 
when  the  vapor  was  saturated. 

On  calculating  the  molecular  volumes  he  found  them  to  be 
from  44  to  72%  of  the  values  calculated  according  to  Kopp's 
rule. 

Schneebeli1  in  1885  attempted  to  get  with  the  greatest  accu- 
racy the  absolute  value  of  the  coefficient  of  friction  of  air  and 
its  dependence  on  the  temperature.  He  used  five  different 
capillaries  and  obtained  the  following  values  of  y  x  107  at  0°  C. : 
1712,  1690,  1698,  1703,  1734,  corrected  for  the  vapor  tension 
of  water.  For  its  variation  with  the  temperature  he  proposed 
the  formula  yt  =  T?O  (1  +0.0027  /)•  His  values  agree  quite  well 
with  those  which  Obermayer  obtained  ten  years  before. 

Koenig2  studied  the  influence  of  magnetization  and  electrifi- 
cation on  the  coefficient  of  friction  and  could  detect  none. 
He  made  an  important  contribution  to  the  subject  of  internal 
friction  by  deducing  a  correction  for  Coulomb's  method  which 
brought  its  results  down  into  agreement  with  those  of  the 
capillary  method. 

Tomlinson3  made  a  very  careful  study  of  the  coefficient  of 
viscosity  of  air  and  its  change  with  the  temperature.  He 
observed  the  logarithmic  decrement  of  the  torsional  vibrations 
of  cylinders  and  spheres.  He  found  T?  at  0°  C.  =  .00017155 
and  thought  Holman's  formula  correct,  j]t  =  i?0  (1  +0.002751  t 
-  0.00000034  t2).  This  result  is  about  9%  lower  than  Maxwell's, 
which  Stokes  explains  on  the  supposition  that  Maxwell's  disks 
were  not  exactly  level.  Tomlinson  also  studied  the  effect  of 
aqueous  vapor  in  the  air.  He  says  that  air  at  15°  C.  and  760  mm. 
pressure  when  saturated  with  aqueous  vapor  is  only  .2%  more 
viscous  than  dry  air.  It  is  only  when  under  a  pressure  less 
than  350  mm.  that  the  aqueous  vapor  begins  to  show  an  ap- 
preciable effect;  but  when  the  rarefaction  is  great,  moist  air 
becomes  considerably  less  viscous  than  dry  air.  Crookes  had 


1.  Archiv.  de  Geneve,  1885,  Vol.  14  (3),  p.  197. 

2.  Wied.  Ann.,  1885,  Bd.  25,  p.  618;  1887,  Bd.  32,  p.  193. 

3.  Phil.  Trans.,  1886,  Vol.  177  (2),  p.  767. 


28 

likewise  said  that  at  15°  C.  and  pressures  760  to  350  mm. the 
presence  of  aqueous  vapor  has  little  or  no  effect  on  the  internal 
friction  of  air. 

Lampel,1  after  reviewing  the  mathematical  treatment  of  the 
torsional  oscillations  of  a  sphere  with  air  resistance  given  by 
Lampe,2  Boltzmann4,  Kirchhoff  and  Klemencic,  proceeded  to 
determine  which  of  these  men  came  nearest  the  truth.  His 
experiments  showed  that  Boltzmann's  formula  is  the  best. 

In  1887  O.  E.  Meyer3  revised  his  mathematical  treatment  of 
Coulomb's  method,  accepting  the  correction  which  Koenig 
had  been  fortunate  enough  to  devise  for  the  disturbance  which 
takes  place  at  the  edges  of  the  cylindrical  surface  of  the  disks. 
He  showed  how  this  correction  improved  his  previous  values 
for  j). 

Because  the  French  physicist,  Hirn,  rejected  the  kinetic 
theory  of  gases  for  the  reason  that  he  could  find  no  increase 
in  the  coefficient  of  friction  of  air  with  increase  of  temperature 
De  Heen5  undertook  an  investigation  of  this  subject.  His 
method  consisted  in  letting  a  brass  piston  descend  under  gravity 
in  a  tube  and  drive  air  through  a  capillary  stopcock  at  the 
bottom.  His  pressures  varied  from  10  to  2280  mm.  of  mercury. 
He  found  y  at  low  pressures  smaller  and  at  high  pressures 
larger  than  at  ordinary  pressures.  He  maintains  that  below 
80  mm.  pressure  r?  varies  as  the  square  root  of  the  absolute 
temperature  in  accordance  with  the  kinetic  theory  of  gases. 
After  passing  80  mm.  pressure  ^  increases  more  rapidly  with 
the  temperature  than  at  lower  pressures,  the*  variability  of  y 
attaining  a  maximum  at  300°  C.  He  thinks  that  the  disagree- 
ment between  theory  and  experiment  may  be  due  to  the  fact 
that  the  mean  free  path  of  a  molecule  is  possibly  not  a  straight 
line  in  gases  which  are  under  the  relatively  high  pressure  of 
.the  atmosphere. 

Couette6,  criticizing  Coulomb's  method  as  giving  only  an  ap- 


1.  Wien.  Ber.,  1886,  Bd.  93  (2),  p.  291. 

2.  Programm  des  Stadt.  Gym.  zu  Danzig,   1866. 

3.  Ann  .d.  Phys.,  1887,  Bd.  32,  p.  642. 

Sitz.  d.  Munch.  Akad.,  1887,  Bd.  17,  p.  343. 

4.  Wien.  Ber.,  1881,  Bd.  84  (2),  p.  40. 

5.  Bull,  de  1'Acad.  de  Belgique,  1888  (3),  Vol.  16,  p.  195. 

6.  Comp.  Rend.,  1888,  Vol.   107,  p.  388. 
Jour,  de  Phys.,  1890  (2),  Vol.  9,  p.  414. 

Ann.  de  Chim.  et  de  Phys.,  1890  (6),  Vol.  21,  p.  433. 


29 

proximation,  even  with  very  slow  oscillations,  and  Poiseuille's 
method,  as  being  true  only  for  very  slow  rates  of  flow,  adopted 
the  method  suggested  by  Margules  in  1881.  He  used  two 
copper  cylinders,  the  inner  suspended  by  a  torsion  thread  and 
the  other  one  coaxial  with  it,  revolving  with  a  constant  velocity. 
The  outer  cylinder  tends  to  set  the  inner  one  in  motion,  which 
is  kept  in  its  primitive  position  by  turning  the  torsion  head 
through  a  measured  angle.  He  found  the  angle  of  torsion 
divided  by  the  revolutions  per  minute  to  be  almost  constant, 
increasing  only  very  slightly  with  the  speed.  His  value  of  T? 
for  air  at  18°  C.  -  .0001847. 

Barus1  made  a  careful  investigation  of  the  viscosity  of  gases 
at  high  temperatures.  His  capillary  was  of  platinum,  radius 
at  0°  C.  =  0.0079,  coiled  into  a  helix.  The  range  of  tem- 
perature was  from  5°  to  1400°  C. 

He  found  that  the  mean  increase  of  gaseous  viscosity  was 
proportional  to  the  \  power  of  the  absolute  temperature  and 
not  in  accordance  with  Schumann's  or  Holman's  formula. 
He  suggested  that  if  the  law  connecting  T?  and  the  temperature 
were  rigorously  known  his  apparatus  could  be  used  as  a  pyro- 
meter. He  pointed  out  that  below  100°  air  is  not  rigorously 
a  perfect  gas,  because  below  that  temperature  its  temperature 
exponent  increases  to  0.73.  Above  100°  C.  it  is  0.67  the  same 
as  for  hydrogen  whose  exponent'  does  not  change  from  0°  to 
1400°  C.  Obermayer  showed  that  as  the  state  of  vapor  is  ap- 
proached the  temperature  exponent  increases,  being  nearly 
1  for  many  vapors.  Barus's  absolute  values  of  y  are  not  good, 
that  for  air  at  0°  C.  being  0.0002472  to  0.0002508,  which  is 
much  too  high. 

Sutherland2  showed  that  the  discrepancy  between  theory 
and  experimental  results  concerning  the.  increase  of  y  with 
the  temperature,  disappears  if,  in  the  theory  account  is  taken 
of  the  forces  of  attraction  between  molecules  when  the  latter 
approach  each  other.  These  attractive  or  cohesive  forces  tend 
to  increase  the  number  of  collisions  between  the  molecules, 
and  thus  have  the  effect  of  apparently  increasing  the  diameter 


1.  Amer.  Journ.  Sci.,  1888  (3),  Vol.  35,  p.  407. 

Bull,  of  U.  S.  Geol.  Survey,  No.  54,  Washington,  1889,  p.  239- 
Wied.  Ann.,   1889,  Bd.  36^  p.  358. 
Phil.  Mag.,  1890,  Vol.  29,  p.  337. 

2.  Phil.  Mag.,   1893  (5),  Vol.  36,  p.  507. 


30 

of  the  sphere  of  action.  With  increased  molecular  velocity, 
due  to  rise  of  temperature,  the  cohesive  forces  have  less  chance 
to  act  than  at  lower  molecular  velocity,  i.e.,  at  lower  tem- 
perature. Therefore  the  sphere  of  action  will  be  increased  less 
at  high  than  at  low  temperatures,  which  produces  the  same 
result  as  if  it  diminished  at  higher  temperatures.  Sutherland 
proposed  the  following  formula  which  agrees  well  with  the  ob- 
servations of  Holman  and  Barus: 


/  T  U       '  273 

f)     =    T)  J  —  I 


in  which  T  is  the  absolute  temperature,  and  C  is  the  cohesion 
constant  depending  upon  the  nature  of  the  gas,  which  for  air 
=  113. 

Ortloff1  made  a  careful  study  of  the  friction  of  the  three 
gases  C2H6,  C2H4  and  C2H2  by  the  transpiration  method.  He 
found  that  the  difference  in  cross  section  of  the  molecules 
C2H6  and  C2H4  is  less  'than  that  between  C2H4  and  C2H2,  the 
relative  values  being  342,  298  and  228.  O.  E.  Meyer2  claims 
that  the  cross  section  of  a  molecule  =  sum  of  the  cross  sections 
of  its  atoms.  Ortloff  found  that  his  molecular  cross  section 
was  less,  his  molecular  diameter  less,  and  his  molecular  volume 
greater  than  those  calculated  according  to  Meyer,  with  the 
exception  of  the  C2H2  volume.  He  concludes  that  his  gas  atoms 
cannot  be  disposed  in  a  plane.  Meyer's  supposition  that  the 
molecular  volume  =  sum  of  the  atomic  volumes  presupposes 
that  the  atoms  are  grouped  in  a  sphere.  Ortloff 's  experiments 
seemed  to  show  that  of  his  three  gases  this  is  true  only  in  the 
case  of  C2H2. 

Noyes  and  Goodwin3  investigated  the  viscosity  of  the  vapor 
of  mercury  because  the  latter  is  a  monatomic  element.  They 
used  the  transpiration  method  and  experimented  also  on 
hydrogen  and  carbon  dioxide.  They  found  that  the  cross 
section  of  a  mercury  molecule  is  2.48  times  as  large  as  that 


1.  Inaug.  Diss.  Jena,  1895. 

2.  Kinetic  Theory  of  Gases,  1899,  Chapter  X. 

3.  Physical  Review,  1896,  Vol.  4,  p.  207. 

Zeitsch.  Physik.  Chem.,  1896,  Vol.  21,  pp.  671-679. 


31 

of  hydrogen  at  300°  C.  They  concluded  that  atoms  and  mole- 
cules are  of  the  same  order  of  magnitude,  and  that  the  spaces 
between  the  atoms  within  the  molecule,  if  any  exist,  are  not 
large  in  comparison  with  those  occupied  by  the  atoms  themselves, 
and  that  therefore  the  coefficient  of  friction  is  not  adapted 
for  distinguishing  between  monatomic  and  polyatomic  mole- 
cules. They  believed  that  this  explains  the  fact  that  the 
molecular  cross  section  of  most  comparatively  simple  molecules 
is  approximately  equal  to  the  sum  of  the  atomic  cross  section 
as  had  been  pointed  out  by  O.  E.  Meyer.  Their  work  is  open 
to  criticism  because  a  joint  in  their  apparatus  could  not  be 
made  air  tight. 

Houdaille1  measured  the  coefficient  of  friction  of  air  and 
vapor  of  water  by  the  transpiration  method.  At  76  cm.  pres- 
sure he  found  i?0  =  0.000186  for  air,  and  for  water  vapor  = 
0.0000975.  At  a  pressure  of  between  1  and  3  cm.  he  found 
f)0  for  air  the  same,  while  r)0  for  water  vapor  changed  to  0.0000885. 
He  found  fair  agreement  between  the  calculated  and  observed 
values  of  the  diffusion  coefficient  for  water  vapor. 

Perot  and  Fabry2  brought  out  a  new  kind  of  absolute  electro- 
meter intended  for  the  measurement  of  small  differences  of 
potential.  They  observed  that  it  attained  its  position  of 
equilibrium  very  slowly  when  the  distance  between  the  plates 
was  small,  owing  to  the  viscosity  of  the  layer  of  air  which 
separates  them.  Thus- they  obtained  a  new  method  for  deter- 
mining the  viscosity  of  air,  which  they  found  to  be  0.000173 
at  13°  C. 

Jaeger3  gave  a  careful  mathematical  analysis  of  the  influence 
of  molecular  volume  on  the  internal  friction  of  gases  taking 
account  of  association  and  expansion  of  molecules.  Instead  of 

the  usual  formula  rt   =  -»-  d  G  L  he  proposed  rt  =  —  d    G  L    in 

which  d  =  density,   G  =  velocity   of  mean   square,   L  =  mean 
free   path  of  a  molecule.       He  also  gave  the  formula   9  =  i)9 
(1+4/3  A)2   .n   whkh   A  ^  l  +    D  ^  +       and   p  =    b_ 
A  2t  V 

of   the  molecular  and  specific  volumes. 


1.  Fortschr.  d.  Phys.,  1896,  52,  Jahr.  I.,  p.  442. 

2.  Compt.  Rend.,   1897,  Vol.  124,  p.  28. 

Ann.  de  Chim.  et  de  Phys.,  1898  (7),  Vol.  13,  p.  275. 

3.  Wien.  Ber.  Mathem.  Naturw.,  1899  (2),  Vol.  108,  p.  447. 

1900  (2),  Vol.  109,  p.  74. 


32 

Breitenbach,1  using  the  transpiration  method,  experimentally 
determined  the  coefficients  of  friction  of  air,  ethylene,  carbon 
dioxide,  methyl  chloride,  and  hydrogen.  He  found  that  T? 
varies  according  to  a  power  of  the  absolute  temperature  whose 
exponent  for  different  gases  varies  between  0.6  and  1.0.  He 
inferred  that  the  sphere  of  action  of  a  molecule  diminishes 
with  increase  of  temperature.  For  the  same  gas  this  exponent 
decreases  with  increasing  temperature.  Also  a  lowering  with 
lower  temperatures  was  noticed.  In  gaseous  mixtures  TJ  does 
not  vary  as  the  composition,  and  Puluj's  formula  is  only  ap- 
proximately correct.  The  difference  between  the  results  of 
the  oscillation  and  transpiration  methods  at  high  temperatures 
cannot  be  explained  by  an  increase  of  the  slipping  of  the  gas 
along  the  capillary  walls.  This  slipping  he  found  could  in 
general  be  neglected. 

Two  years  later  Breitenbach2  compared  his  work  with  Suther- 
land's formula  for  the  variation  of  r?  with  the  temperature,  and 
found  this  formula  gave  very  excellent  agreement  with  his 
experiments.  The  value  of  the  cohesion  constant  for  air  he 
found  to  be  119.4  instead  of  113  according  to  Sutherland. 

Rayleigh3  on  the  supposition  that  jjt  =  T?O  (1  +a  t)n  found 
the  exponent  n  for  dry  air  =  0.754,  for  oxygen  =  0.782,  for 
hydrogen  0.681,  for  argon  (impure)  0.801,  for  pure  argon  = 
0.815.  Later,  using  Sutherland's  formula,  he  found  the  cohesion 
constant  to  be  the  same  for  hydrogen  and  helium,  viz.:  72.2, 
and  this  value  and  those  for  air,  oxygen  and  argon  agree  well 
with  the  values  calculated  by  Sutherland  from  Obermayer's 
observations. 

Schultze4,  like  Rayleigh,  used  the  transpiration  method  for 
investigating  the  internal  friction  of  argon  and  its  change  with 
the  temperature.  He  found  its  friction  at  0°  C.  to  be  2104  X  10-7 
and  that  it  varied  with  the  temperature  in  accordance  with 
Sutherland's  formula,  the  cohesion  constant  being  169.9.  His 
argon  contained  J%.  of  nitrogen. 

Job5  called  attention  to  a  new  method  of  measuring  the  re- 
sistance offered  by  a  capillary  tube  to  the  flow  of  gases.  A 
voltameter  is  provided  with  a  capillary  outlet;  the  pressure 


1.  Wied.  Ann.,  1899,  Bd.  67,  p.  803. 

2.  Drude's  Ann.,   1901,  Bd.  5,  p.   166. 

3.  Proc.  Roy.  Soc.,  1900,  Vol.  66,  p.  68;  Vol.  67,  p.  137. 

4.  Drude's  Ann.,   1901,  Bd.  5,  p.   140. 

5.  Bull.  Soc.  Franc.  Phys.,  1901,  Vol.   157,  p.  2. 


33 

produced  in  the  voltameter  for  a  given  current,  measures  the 
resistance  of  the  capillary.  The  author  suggests  the  applica- 
cation  of  this  method  to  various  experiments  in  connection  with 
the  flow  of  gases. 

F.  G.  Reynolds,1  using  spheres  and  cylinders  in  torsional 
oscillation,  determined  the  viscosity  coefficient  of  air,  and 
investigated  the  effect  upon  it  of  Rontgen  rays.  His  value  of  y 
at  21°  C.  =  187  X 10-6.  The  results  of  his  experiments  with 
Rontgen  rays  seem  to  show  that  their  effect  is  scarcely,  if  at  all 
perceptible. 

Bestelmeyer,2  using  transpiration  apparatus  very  similar  to 
Holman's,  investigated  the  change  in  the  coefficient  of  internal 
friction  of  nitrogen  for  a  temperature  range  of— 192°  C.  to 
+  300°  C.  and  found  that  Sutherland's  formula  applied  with 
satisfactory  accuracy  except  at  —  192°  where  it  was  2%  in 
error,  perhaps  owing  to  change  in  the  law  at  this  low  tempera- 
ture. The  cohesion  constant  was  found  to  be  110.6. 

Markowski,3  employing  the  same  transpiration  apparatus 
used  by  Schultze  in  1901,  studied  the  internal  frictions  of 
oxygen,  hydrogen,  chemical  and  atmospheric  nitrogen,  and 
their  change  with  the  temperature.  He  corrected  all  his  results 
for  slip,  using  coefficients  of  slip  determined  by  Breitenbach 
Kundt  and  Warburg,  for  air,  oxygen  and  hydrogen.  For 
nitrogen  he  used  the  molecular  free  path,  which  is  theoretically 
nearly  equal  to  the  coefficient  of  slip.4  His  corrected  value  of 
TI  for' air  at  15.9°  C.  -  1814  xlO-7,  and  at  99.62°=  2212  XlO-7. 
The  correction  for  slip  amounted  to  J%.  He  states  that  Graham's 
results  are  preferable  to  those  of  Obermayer.  Sutherland's 
formula  he  found  to  give  excellent  results  from  0°  to  183°  C. 
He  also  tried  the  older  exponential  formula  and  found  the  ex- 
ponent diminishes  with  rise  of  temperature  as  had  been  noticed 
by  several  other  observers.  Puluj's  formula  for  the  friction 
of  a  gaseous  mixture  he  found  applies  well  to  atmospheric 
nitrogen  as  a  mixture  of  chemically  pure  nitrogen  and  argon. 

Kleint,5    continuing    the    work    of    Schultze    and  Markowski 


1.  Phys.  Review,  1904,  Vol.  18,  p.  419;  Vol.  19,  p.  37. 

2.  Inaug.  Diss.  Munich,  1902. 

Ann.  d.  Phys.,  1904,  Bd.  13,  p.  944. 

3.  Inaug.  Diss.  Halle,  1903. 

Ann.  d.   Phys.,   1904,  Vol.   14,  p.  742. 

4.  O.  E.  Meyer's  Kinetic  Theory  of  Gases,  1899,  p.  211. 
o.   Inaug.  Diss.  Halle,  1904. 


34 

with  the  same  piece  of  apparatus  investigated  the  friction  of 
mixtures  of  oxygen,  hydrogen  and  nitrogen.  He  found  Puluj's 
formula  to  give  only  approximately  the  coefficient  of  friction 
of  a  mixture  of  gases.  He  observed  that  small  percentages 
of  oxygen  and  nitrogen  raise  the  friction  of  hydrogen  markedly, 
while  hydrogen  only  begins  to  make  itself  felt  upon  the  others 
when  it  is  5%  of  the  mixture.  Sutherland's  formula  for  the 
increase  of  y  with  the  temperature  he  found  to  be  correct. 

METHOD  EMPL.OYED  IN  EXPERIMENTAL  INVESTIGATION. 
The  preceding  historical  review  shows  that  the  methods 
employed  in  the  past  for  determining  the  coefficient  of  the 
internal  friction  of  gases,  may  be  divided  into  two  general 
classes:  those  in  which  the  movement  of  a  solid  body  in  the 
gas  is  observed,  and  those  in  which  the  time  of  passage  of  the 
gas  through  a  capillary  tube  is  noted.  There  are  many  modifi- 
cations of  both  general  methods,  especially  of  the  first  named 
one  which  is  the  earlier  historically.  Owing  to  the  difficulty 
of  getting  capillaries  of  perfectly  uniform  bore,  and  of  deter- 
mining with  the  greatest  accuracy  the  shape  and  size  of  their 
cross  section,  and  also  owing  to  the  possible  formation  of  eddies 
at  the  beginning  and  end  of  the  flow,  and  to  the  possible  slipping 
of  the  gas  on  the  capillary  walls,  probably  the  oscillation  method 
is  the  better  for  absolute  measurements,  provided  a  solid  body  is 
employed  of  such  a  shape  that  the  mathematical  treatment 
is  rigorously  correct.  It  seems  certain  however  that  the 
transpiration  method  has  been  growing  in  favor  of  late  years, 
and  that  it  is  more  convenient  for  comparative  measurements 
than  the  other  method.  As  already  stated  my  object  was  more 
to  get  comparative  than  absolute  values  of  the  greatest  accu- 
racy, hence  the  transpiration  method  was  decided  upon  for 
this  experimental  investigation. 

GENERAL  DESCRIPTION  OF  APPARATUS. 

The  form  of  apparatus  and  the  subject  of  this  research  were 
kindly  suggested  to  me  by  Professor  Morris  Loeb,  under  whose 
guidance  these  experiments  have  been  made.  The  apparatus 
consists  essentially  of  a  U  shaped  tube  (see  Fig.  1),  one  limb 
of  which  is  capillary,  while  the  other  is  not,  but  serves  as  a 
cylinder  of  known  capacity,  down  which  is  forced  by  gravity 
a  piston  consisting  of  a  column  of  mercury,  which  drives  the 
gas  under  it  up  through  the  capillary  limb.  The  capillary,  At 


T 


M, 


B 


FIG.  1. 

View  perpendicular  to  the 
plane  of  the  two  limbs 


v 


FIG.  2. 

View  in  the  plane  of 
the  limbs. 


fo. 

FIG.  3. 
Steam  Jacket. 


36 

does  not  begin  at  the  bend  of  the  U,  but  some  distance  above 
it.  The  distance  between  the  limbs  is  only  3  cm.,  so  that  the 
apparatus  can  be  placed  inside  of  a  glass  tube  5  cm.  in  diameter 
which  serves  as  a  steam  jacket  (see  Fig.  3),  the  steam  being 
introduced  through  a  side  tube,  S,  near  the  top  and  passing 
out  freely  at  the  bottom,  0.  A  cork,  C,  through  which  both 
limbs  of  the  apparatus  pass  is  on  a  level  with  the  top  of  the 
capillary,  and  serves  to  close  the  steam  jacket  at  the  top. 
The  mercury  limb  above  the  cork  is  provided  with  a  special 
stop-cock,  H,  bored  out  to  the  same  diameter  as  the  tube  below 
it.  Above  the  stop-cock  is  a  continuation  of  the  tube  ter- 
minating in  a  small  funnel,  F,  for  convenience  in  introducing 
ether  and  mercury.  An  ordinary  thistle  funnel  with  horizontal 
bottom  will  not  answer;  the  bottom  of  the  funnel  must  be 
inclined  about  45°  to  the  axis  of  the  mercury  tube,  otherwise  the 
mercury  will  not  so  readily  descend  in  one  unbroken  column 
when  released  by  turning  the  stop-cock.  Two  marks,  M1  and  M2 
are  etched  on  the  mercury  tube  a  convenient  distance  apart, 
the  lower  one,  M2,  being  near  the  bend  at  the  bottom,  the 
volume  of  the  tube  between  the  marks  being  accurately  deter- 
mined by  calibration  with  mercury.  The  bore  of  this  tube 
being  slightly  conical  instead  of  truly  cylindrical,  its  calibration 
was  also  carried  far  enough  above  the  upper  mark,  M\,  to  cover 
the  space  passed  over  by  the  mercury  column  during  an  ex- 
periment, so  that  the  average  height  of  the  mercury  column 
during  an  experiment  could  be  calculated. 

The  method  of  making  an  observation  on  ether  gas  is  essen- 
tially as  follows:  The  steam  jacket  being  cold  and  filled  with 
air  the  apparatus  is  placed  in  it.  The  stop-cock  is  opened 
and  about  1  cc.  of  ether  is  poured  down  the  mercury  tube. 
By  means  of  an  aspirator  the  ether  is  sucked  up  the  other  limb 
close  to  the  base  of  the  capillary.  The  small  hermetically 
closed  tube  seen  at  V  in  Figs.  1  and  2  serves  as  a  reservoir  for 
ether  at  the  base  of  the  capillary.  Care  was  always  taken  to 
fill  this  tube  completely  with  ether  so  that  it  could  not  act 
as  an  air  pocket.  Steam  is  then  led  into  the  jacket  and 
of  course  vaporizes  the  ether,  driving  surplus  liquid 
violently  out  through  the  top  of  the  mercury  tube.  The 
stop-cock  is  then  closed  and  after  equilibrium  has  been  estab- 
lished, a  weighed  amount  of  mercury  is  introduced  into  the 
funnel  and  allowed  to  flow  down  until  it  is  arrested  by  the 
stop-cock,  which  is  not  quite  air  tight,  because  no  grease  can 


37 

be  used  in  it  for  fear  of  soiling  the  mercury.  The  stop-cock  is 
then  suddenly  turned  and  the  mercury  descends  in  a  solid 
column.  When  its  lower  meniscus  passes  the  upper  mark  on 
the  tube  a  stop  watch  is  started,  which  is  stopped  as  the  same 
meniscus  passes  the  lower  mark.  The  barometer  is  read  when 
the  mercury  has  covered  half  the  measured  distance.  A  ther- 
mometer, T,  Fig.  1,  hanging  from  the  cork  at  the  top  of  the 
steam  jacket  indicates  the  temperature  of  the  steam',  and  serves 
as  a  plumb  line.  The  readings  of  this  thermometer  were 
corrected  by  comparison  with  a  standard  thermometer.  The 
coefficient  of  friction  of  the  gas  is  calculated  by  the  following 
formula  given  by  O.  E.  Meyer:1 

xgdr 
1GL  V 

in  which  d    =  density  of  mercury  at  0°  C 

g    =  acceleration  due  to  gravitation 

r    =  radius  of  capillary  in  cm. 

L  =  length  of  capillary  in  cm. 

P  =  pressure  of  gas  on  entering  the  capillary. 

p    =  height  of  barometer  at  0°  C 

t     =  time  in  seconds 

V  =  c.c.  of  gas  transpired. 

No  allowance  need  be  made  for  the  expansion  of  the  mercury 
at  100°  C.,  for  what  the  column  gains  in  height  is  compensated 
for  by  loss  in  density.  No  allowance  has  been  made  for  the 
expansion  of  the  capillary  as  it  was  assumed  the  expansion 
of  the  mercury  tube,  hence  increase  of  the  volume  of  gas  trans- 
pired, would  counteract  this. 

The  advantages  of  this  apparatus  over  that  devised  by  L. 
Meyer  are  quite  obvious  and  numerous.  In  the  first  place  it 
is  much  more  simple  and  available.  The  capillary  is  straight 
instead  of  coiled  into  a  helix.  The  friction  of  the  vapors  is 
taken  at  a  temperature  so  high  above  the  boiling  point  that 
they  behave  more  like  true  gases  than  at  their  boiling  points, 
where  they  are  in  a  condition  of  unstable  equilibrium.  The 
length  of  time  required  for  an  experiment  is  short  enough  for 
all  conditions  to  be  kept  constant,  and  yet  long  enough,  so 
that  with  a  stop  watch  reading  to  one  fifth  of  a  second,  the 
time  can  be  determined  to  within  less  than  one  part  in  a  thousand. 


Pogg.  Ann.,   1866.  Bd.   127,  p.   269. 


38 

APPARATUS  No.  1. 

The  first  piece  of  apparatus  was  constructed  more  to  test 
the  practicability  of  the  method  than  for  getting  accurate 
results.  It  having  been  determined  by  experiment,  that  the 
largest  diameter  of  tube  in  which  a  mercury  column  would 
hold  together,  against  a  cushion  of  air,  was  about  .35  cm.,  this 
apparatus  was  constructed  with  a  mercury  tube  whose  mean 
diameter,  where  it  was  traversed  by  the  mercury  column  was 
.35104  cm.,  as  shown  by  the  fact  that  76.0405  grams  of  mercury 
occupied  58.05  cm.  at  20°  C.  The  distance  between  the  two 
etched  marks  on  the  tube  was  50  cm.  This  distance  was  occu- 
pied by  65.5558  grams  of  mercury  at  20°  C.  The  density  of 
mercury  at  this  temperature  being  13.5463  the  volume  between 
the  two  marks  was  4.8394  cubic  centimetres. 

The  capillary  was  34.35  cm.  in  length  which  was  only  about 
half  the  length  of  the  mercury  tube  forming  the  other  side 
of  the  U,  in  other  words  the  capillary  formed  the  upper  half 
of  one  side  of  the  U.  Its  bore  was  conical  as  shown  by  the 
fact  that  9.50  cm.  of  mercury  at  the  small  end  occupied  9.05  cm. 
at  the  larger  end  which  was  made  the  bottom  because  O.  Rey- 
nolds1 has  shown  that  converging  walls  tend  to  a  steady  instead 
of  a  whirling  flow  of  water  and  presumably  also  of  gas.  The 
above  column  of  mercury  was  passed  slowly  from  one  end  of 
the  capillary  to  the  other  and  changed  its  length  gradually 
and  steadily,  thus  showing  the  absence  of  abnormal  contrac- 
tions or  expansions  of  the  bore.  The  bore  of  the  capillary 
was  determined  from  a  sample  5.75  cm.  long  which  had  been 
cut  from  the  smaller  end.  This  sample  was  weighed  several 
times  empty  and  when  containing  different  columns  of  mercury 
and  the  radius  of  the  bore  determined  in  the  well  known  way. 
A  section  was  also  examined  under  a  microscope  with  a  micro- 
meter eye  piece,  and  found  to  be  so  nearly  a  true  circle  that  it 
was  taken  for  such.  The  microscope  reading  agreed  well  with 
the  mercury  determinations  of  the  radius.  After  taking  the 
average  of  the  mercury  and  microscope  readings  and  after 
allowing  for  the  taper  of  the  capillary  its  mean  radius  was  found 
to  be  .0103908  cm. 

To  facilitate  the  washing  of  the  apparatus  the  small  vent, 
V,  in  Figs.  1  and  2,  was  left  about  \  cm.  below  the  base  of  the 
capillary.  This  vent  tube  which  was  only  1  cm.  long  was 


1.   Proc.  Royal  Inst.  Grt.  Brit.,  1884,  Vol.  11,  p.  44. 


39 


hermetically  sealed  off  after  the  apparatus  was  washed  and 
ready  for  use.  The  apparatus,  after  it  was  received  from  the 
glass  blower,  was  washed  throughout  with  a  solution  of  potassium 
permanganate  made  alkaline  with  caustic  potash,  then  with  a 
solution  of  the  same  substance  made  acid  with  dilute  sulphuric 
acid.  Later  a  half  and  half  solution  of  potassium  bichromate 
and  strong  sulphuric  acid  was  substituted  for  the  above.  Some- 
times also  strong  nitric  acid,  followed  by  dilute  nitric  acid 
was  used.  Then  followed  many  washings  with  distilled  water. 
The  apparatus  was  then  dried,  by  sucking  through  it  by  means 
of  an  aspirator,  hot  air  which  was  filtered  through  cotton  and 
dried  by  calcium  chloride.  A  column  of  calcium  chloride  was 
also  placed  in  the  connection  between  the  aspirator  and  the 
apparatus.  The  drying  was  not  hastened  by  the  use  of  alcohol 
or  ether.  The  mercury  was  cleansed  by  shaking  with  dilute 
nitric  acid,  passing  through  a  fine  pin  hole  and  drying  in  a  porce- 
lain dish  at  110°  C. 


TABLE  OF  THE  INTERNAL  FRICTION  OF  AIR. 

APPARATUS  No.  1. 
Ht.  of  Press,  of 

Temp,     driving    Barom.    entering  Time  in     T^XlO7  Average 

in  C°         Col.        atO°C        Gas       seconds 
in  cm. 


15 

7.97 

76.03 

83.09 

76.04 

1872 

14.9 

7.97 

76.03 

83.09 

75.0 

1862 

14.9 

7.97 

76.03 

83.09 

74.8 

1857 

1863.  8±  2.  57 

14.9 

7.966 

76.03 

83.09 

75.2 

1866 

14.9 

7.966 

76.60 

83.64 

75.2 

1860 

15.4 

7.966 

76.60 

83.64 

75.4 

1865 

20.4 

4.29 

75.38 

78.87 

156.8 

1967 

1972±3.5 

20.2 

4.29 

75.38 

78.87' 

157.6 

1977 

21.4 

9.48 

75.38 

84.01 

64.4 

1935 

1932±2.0 

21.4 

9.48 

75.38 

84.01 

64.2 

1929 

99.9 

9.48 

75.38 

84.01 

75.6 

2272. 

5 

99.9 

9.48 

75.38 

84.01 

76.0 

2284. 

5 

100.3 

7.98 

76.96 

84.00 

92.0 

2238. 

2 

100.3 

7.98 

76.96 

84.00 

89.8 

2185. 

0 

2296.  5±  15  8 

100.3 

7.97 

76.60 

83.64 

94.0 

2305. 

0 

100.3 

7.97 

76.60 

83.64 

96.4 

2384. 

5 

100.3 

7.97 

76.60 

83.64 

92.7 

2293. 

0 

100.3 

7.97 

76.60 

83.64 

94.0 

2325. 

0 

40 

The  results  of  experiments  on  air  with  this  apparatus  are 
given  in  the  preceding  table.  It  will  be  noticed  they  are  all  rather 
high,  but  prove  at  any  rate  that  the  method  and  apparatus  are 
practicable.  The  probable  errors  were  calculated  by  the  usual 

formula  0.6745  J    ^ dlfp       The   error  for  readings  at   100°  C. 
\    n(n~l] 

is  large,  probably  owing  to  an  error  of  the  stop  watch,  whose 
hand  showed  a  tendency  to  fly  forward  when  stopping  near 
90  seconds.  In  one  case  the  hand  flew  forward  10  seconds. 
The  watch  was  of  course  repaired  as  soon  as  this  defect  was 
noticed.  In  those  cases  where  the  mercury  was  allowed  to 
run  back  and  was  used  over  again  the  readings  were  weighted 
less  than  independent  readings,  three  readings  being  con- 
verted into  two  by  taking  their  means,  and  four  into  three  in 
the  same  way. 

It  is  rather  difficult  to  tell  what  is  the  true  coefficient  of 
the  internal  friction  of  air.  Landolt  and  Boernstein1  give  r) 
for  air  at  15°  C.  1784  X  107.  Markowski2  at  16°  gives  1814  X  107, 
Kleint3  at  14.1  to  14.5°  C.  gives  1808  X  107.  According  to  Landolt 
and  Boernstein  then,  my  readings  are  4J%  too  high  at  15°, 
while  according  to  Markowski  &  Kleint  they  are  about  3%  too 
high.  F.  G.  Reynolds4  gives  for  air  at  20.7°  .000187  which  is 
3J%  lower  than  my  result  of  .0001932  at  21.4°  C. 

It  was  found  by  repeated  trials  with  di-ethyl  ether,  that 
the'  surface  tension  of  the  mercury  was  so  much  reduced  by 
contact  with  the  ether  gas,  that  the  mercury  would  not  hold 
together  in  one  column  in  apparatus  No.  1.  It  was  determined 
by  experiment  that  the  mercury  piston  could  not  be  used  for 
ether  gas  in  a  tube  whose  diameter  was  much  larger  than  2  mm. 
A  second  piece  of  apparatus  was  accordingly  constructed  with 
a  mercury  tube  of  about  this  size,  and  a  finer  capillary,  so 
that  the  transpiration  time  would  be  increased  and  greater 
accuracy  be  secured. 

APPARATUS  No.  2. 

The  capillary  selected  for  this  when  examined  with  a  simple 
microscope  was  at  first  thought  to  be  circular  in  cross  section, 


1.  Phys.  Chem.  Tabellen,  Landolt  &  Boernstein,   1893. 

2.  Inaug.  Diss.  Halle,  1903. 

3.  Inaug.  Diss.  Halle,  1904. 

4.  Physical  Rev.,  1904,  Vol.  18,  p.  419;  Vol.  19,  p.  37. 


41 

but  the  use  of  a  high  power  microscope  with,  micrometer 
eye  piece  showed  it  to  be  very  elliptical,  the  ratio  of  the  axes 
being  almost  exactly  as  3  is  to  1.  The  area  of  its  cross  section 
was  determined  by  mercury  several  times  and  these  values 
averaged  with  the  microscope  reading  with  which  they  agreed 
well. 

The  capillary  which  was  84  cm.  long  tapered  in  its  bore, 
2.5  cm.  of  mercury  at  the  large  end  becoming  2.685  cm.  at  the 
small  end.  The  sample  whose  bore  was  determined  was  taken 
from  the  small  end.  After  allowing  for  the  taper  of  the  bore 
the  average  semi-major  axis  was  found  to  be  .006057  cm.  and 
the  average  semi -minor  axis  .002016  cm.  The  capillary  was 
placed  with  its  larger  end  downward  and  reached  to  within 
15  cm.  of  the  bend  in  the  U. 

The  mean  diameter  of  the  mercury  tube  in  that  part  traversed 
by  the  mercury  piston  during  an  experiment  was  .2012  cm.  as 
shown  by  the  fact  that  31.3  grams  of  mercury  occupied  a 
length  of  72.6  cm.  at  16°  C.  The  marks  on  the  mercury  tube 
were  made  50  cm.  apart,  the  volume  between  these  marks  being 
found  by  means  of  mercury  to  be  1.5707  c.c. 

Because  the  capillary  was  elliptical  the  formula  used  for 
calculating  the  internal  friction  of  gas  from  this  apparatus  is 

TT  dg          a3b3        P2-p2 
"  8  L  V     a2    +  b2  P 

where  a  =  semi  major  axis  of  ellipse  of  capillary 

b  =  semi  minor  axis  of  ellipse  of  capillary 

d  =  density  of  mercury  at  0°  C 

g  =  acceleration  due  to  gravitation 

L  =  length  of  capillary 

V  -•=  vol.  of  gas  transpired 

P  =  pressure  of  entering  gas 

p  =  pressure  of  leaving  gas  =  barom.  at  0°  C. 

t  =  time  in  seconds. 

The  results  with  air  with  this  apparatus  are  shown  in  the  next 
table. 

It  will  be  noticed  that  the  values  of  TJ  are  somewhat  lower 
than  for  apparatus  No.  1,  and  therefore  nearer  the  correct 
values. 

The  different  lengths  of  the  capillary  given  in  the  first  column 
are  due  to  the  fact  that  on  several  occasions  the  upper  end  of 


42 


the  capillary  became  stopped  with  dust  from  the  atmosphere 
and  had  to  be  cut  off.  When  not  in  use  the  capillaries  were 
kept  capped  with  rubber,  the  funnels  filled  with  cotton  and 
closed  by  corks  in  which  were  inserted  tubes  of  chloride  of 
calcium. 

TABLE  OF  THE  INTERNAL  FRICTION  OF  AIR. 

APPARATUS  No.  2. 
Ht.  of  Pres.  of 

Length  Temp,  driving  Barom. entering  Time  in     f)X  107       Average 
of  cap.   in  C°       Col.      at  0°  C      Gas     seconds  >?X107 


81.8 

12.5 

23. 

17 

75. 

74 

97. 

92 

2604 

1864 

.2 

84. 

18.5 

23. 

17 

76. 

19 

98. 

30 

2721. 

1897 

.4 

84. 

19.9 

23. 

15 

76. 

26 

98. 

41 

2715. 

1892 

.0  )  1897.3±3.5 

84. 

19.9 

23. 

42 

76. 

41 

98. 

82 

2700. 

1902 

.5f 

84. 

20.9 

23. 

15 

76. 

26 

98. 

41 

2730. 

1902 

.2 

84. 

26.9 

23. 

16 

76. 

15 

98. 

31 

2810. 

1957 

.1 

84. 

29.1 

23. 

17 

75. 

90 

98. 

08 

2830. 

1973 

.6 

84. 

100. 

23. 

17 

75. 

91 

98. 

09 

3201. 

2233 

.0  ) 

81.8 

99.9 

23. 

14 

75. 

70 

98. 

85 

3130. 

2237 

81.8 

100.2 

23. 

14 

76. 

51 

98. 

65 

3155. 

2257 

.0  ) 

TABLE  OF  THE  INTERNAL  FRICTION  OF  AIR. 
APPARATUS  No.  3. 


73.25 

10. 

18 

.746 

76.78 

94 

.56 

509. 

8 

1843.8 

73.25 

13. 

3 

18 

.75 

76.78 

94 

.56 

516. 

1866.9 

73.25 

14. 

4 

18 

.74 

76.80 

94 

.58 

518. 

2 

1870.0 

73.25 

19. 

4 

18 

.75 

76.00 

93 

.79 

524. 

6 

1894.0 

73.25 

20. 

8 

18 

.74 

75.99 

93 

.78 

531. 

2 

1917.2 

70. 

21. 

0 

18 

.75 

76.99 

94 

.77 

509. 

2 

1923.6 

1909.  9±  7.  2 

70. 

20. 

8 

18 

.75 

76.95 

94 

.74 

500. 

0 

1889.0 

70. 

100. 

18 

.74 

75.76 

94 

.50 

592. 

8 

2244.0 

70. 

100. 

1 

18 

.74 

76.19 

94 

.93 

599. 

2267.0 

2246.  3±7.  2 

70. 

100. 

3 

18 

.75 

77.13 

94 

.92 

588. 

8 

2228.0 

PREPARATION  OF  ETHERS. 

The  di-ethyl  ether  used  in  the  following  experiments  was 
sulphuric  ether,  U.  S.  P.  purified  by  two  washings  with  con- 
centrated sulphuric  acid,  C.  P.,  each  of  these  washings  being 
followed  by  one  with  distilled  water.  It  was  then  shaken  with 
mercury  and  dried  with  sodium  wire  from  which  it  was  dis- 
tilled into  glass  tubes  which  were  afterward  hermetically  sealed. 


43 


TABLE  OP  INTERNAL  FRICTION  OF  ETHER  GASES. 
APPARATUS  No.  2. 


Ht.  of  Press,  of 

Length       Kind      Temp,  driving  Barom.   entering  Time  in 

Capillary  of  Gas.     of  Gas      Col.  at  0°  C      Gas         seconds 

atO°  C 


Average 


81.8 

Methyl- 

99. 

9 

23.16 

75.68 

97 

.85 

1502.0 

1074 

.5 

81.8 

Ethyl 

100. 

23.20 

76.09 

98 

.30 

1537.8 

1103 

81.8 

Ether 

99. 

8 

23.19 

75.87 

98 

.07 

1471.0 

1053 

.9 

81.8 

99. 

8 

23.16 

75.44 

97 

.61 

1558.0 

1114 

.2 

1092.3-^-6.4 

81.8 

100. 

23.06 

76.28 

98.35 

1550.2 

1105 

.4 

81.8 

100. 

23.16 

76.15 

98 

.32 

1540.0 

1102.8 

84. 

Methyl- 

100. 

23.90 

76.28 

99 

.18 

1380.0 

990 

9 

84. 

Propyl 

100. 

23.22 

76.23 

98 

.46 

1409.6 

985 

.4 

84. 

Ether 

100. 

23.28 

76.18 

98 

.47 

1402.2 

982 

.5 

84. 

100. 

1 

23.14 

76.46 

98 

.60 

1461.8 

1018 

.9 

1004.3-^5.5 

82.5 

99. 

e 

23.13 

75.31 

97 

.46 

1440.4 

1021 

.2 

81.8 

99. 

8 

23.11 

74.24 

96 

.39 

1439.0 

1027 

.0 

81.8 

Methyl- 

100. 

8 

23.15 

77.15 

99 

.30 

1481.4 

1061 

.1 

81.8 

Isopro- 

100. 

a 

23.14 

77.10 

99 

.24 

1526.0 

1081 

.1 

81.8 

pyl 

100. 

23.13 

76.01 

98 

.24 

1459.6 

1047 

.3 

1054.8-J-5.0 

81.8 

Ether 

100. 

23.12 

76.00 

98 

.13 

1455.0 

1039 

.9 

81.8 

100. 

23.14 

76.00 

98 

.15 

1460.8 

1044 

.9 

84. 

Di-Eth- 

100. 

i 

23.42 

76.39 

98 

.81 

1410.0 

993 

.4 

84. 

yl  Eth- 

100. 

23.42 

76.08 

98 

.51 

1404.0 

990 

.6 

84. 

er 

100. 

23.16 

76.15 

98 

.32 

1416.6 

986 

.8 

84. 

100. 

23.17 

76.10 

98 

.28 

1417.6 

988 

.6 

999.    -(-3.3 

84. 

100. 

i 

23.16 

76.54 

98 

.71 

1437.6 

1000 

.6 

82.5 

99. 

9 

23.16 

75.71 

97 

.89 

1434.6 

1017 

.8 

82.5 

99. 

8 

23.21 

75.50 

97 

.73 

1428.0 

1015 

.2 

84. 

Ethyl- 

100. 

23.17 

76.06 

98 

.24 

1285.0 

896.2 

84. 

Propyl 

100. 

23.17 

75.89 

98 

.07 

1296.4 

904 

.6 

84. 

Ether 

99. 

D 

23.16 

75.70 

97 

.87 

1281.2 

892 

.7 

81.5 

99. 

I 

23.15 

75.58 

97 

.75 

1295.8 

930 

.5 

91o.2-J.-5.  3 

81.5 

99. 

8 

23.14 

75.57 

97 

.73 

1298.8 

932.3 

81.5 

99. 

8 

23.15 

75.57 

97 

.74 

1300.8 

934 

.0 

The  ethyl-propyl  ether,  methyl-propyl  ether  and  methyl-ethyl 
ether  used  in  this  investigation  were  prepared  by  me  in  the 
research  laboratory  of  New  York  University  under  the  per- 
sonal supervision  of  Professor  Loeb,  according  to  the  continuous 
etherification  method  described  by  Norton  and  Prescott  in  the 
Am.  Chem.  Journal,  1884,  Vol.  VI.,  p.  241.  They  were  care- 
fully dried  with  sodium  and  kept  in  sealed  tubes. 

The  other  substances,  viz.:  di-methyl  ether,  ethyl  alcohol, 
methyl-isopropyl  ether,  ethyl-isopropyl  ether,  di-propyl  ether, 
isopropyl-propyl  ether  and  di-isopropyl,  I  owe  entirely  to  the 
great  courtesy  of  Professor  Loeb,  as  I  had  no  part  at  all  in  their 
preparation.  As  far  as  is  known  this  is  the  first  time  that 


44 

methyl-isopropyl  ether  has  ever  been  made,  while  ethyl-iso- 
propyl  ether  and  isopropyl-propyl  ether  have  probably  been 
made  only  once  before. 

The  internal  frictions  of  the  ethers  used  in  apparatus  No.  2 
are  given  in  the  foregoing  table  on  page  43. 

The  ethers  are  arranged  in  this  table  in  the  order  of 
theii  molecular  weights;  that  of  methyl-ethyl  ether  being 
60.064,  that  of  methyl-propyl,  methyl-isopropyl,  and  di-ethyl 
ether  being  alike  74.08;  while  that  of  ethyl  propyl  is  88.096. 
It  will  be  noticed  the  smaller  the  molecular  weight  the  greater 
is  the  internal  friction,  which  agrees  well  with  the  kinetic  theory 
of  gases,  according  to  which  the  friction  increases  with  dimin- 
ished size  of  molecule. 

The  most  noteworthy  fact  shown  in  this  table  is  that  the 
three  isomeric  ethers,  di-ethyl,  methyl-propyl  and  methyl- 
isopropyl  have  not  the  same  internal  friction.  Di-ethyl  ether 
and  methyl-propyl  practically  agree,  while  methyl-isopropyl 
ether  has  a  friction  5%  higher.  According  to  the  universally 
accepted  view  of  the  concatenation  of  the  molecules  of  these 
substances,  the  first  two  are  simple  chain  compounds,  while 
the  last  is  a  chain  compound  with  two  branches: 

Di-ethyl  ether  H3C— C— O— C— CH3 

H2          H2 

Methyl-propyl  ether        H3C— O— C— C— CH3 

H2  H2 

Methyl-isopropyl  ether  H3C— O— C<£^ 

H       Hs 

It  is  easily  conceivable  how  the  last  arrangement  results  in 
a  compacter,  smaller  molecule,  and  hence  shows  a  higher  in- 
ternal friction. 

Although  the  amount  of  liquid  used  varied  from  ^  cc.  to  1  cc. 
as  a  rule,  this  small  amount  seemed  sufficient  on  vaporizing 
to  drive  all  the  air  out  of  the  apparatus,  as  filling  the  appar- 
atus entirely  with  ether  from  the  base  of  the  capillary  to  the 
stop-cock  did  not  give  results  different  from  those  obtained 
using  1  cc.  of  ether. 

The  weight  of  mercury  used  in  apparatus  No.  1  and  No.  2 
was  usually  about  10  grams.  That  in  apparatus  No.  3  shortly 
to  be  described  was  about  5  grams. 

The  allowance  to  be  made  for  the  friction  of  the  mercury 
against  the  walls  of  the  tube  was  determined  in  the  following 


45 

manner.  Three  readings  with  air  were  taken  with  three  different 
columns  of  mercury,  under  constant  temperature  conditions 
It  being  known  that  the  internal  friction  of  a  gas  is  independent 
of  the  pressure,  the  values  of  y  from  these  three  readings  were 
equated  to  each  other  in  pairs,  giving,  after  cancelling  out 
constants  of  the  apparatus,  equations  of  the  form 

P2c2_p2  P2c2_p2 

PC  P,c         tl 

in  which  c  is  a  constant  reduction  factor  due  to  the  friction  of 
the  mercury  against  the  walls  of  the  tube.  The  effect  of  cap- 
illarity is  probably  negligible,  because  the  upper  and  lower 
surfaces  of  the  mercury  are  convex  in  opposite  directions. 

In  solving  the  equations  for  c  good  agreement  was  found, 
the  average  c  for  apparatus  No.  1  being  .9892.  For  apparatus 
No.  2  c  =  .9895  and  for  apparatus  No.  3  c  =  .9894,  all  at  100°  C. 
The  pressure  of  the  gas  on  entering  the  capillary  was  calculated 
by  adding  the  height  of  the  mercury  column  at  0°  C.  to  the 
barometer  at  0°  C.  and  multiplying  by  the  reduction  factor 
given  above.  The  mercury  and  barometer  column  were  re- 
duced to  0°  C.  by  the  aid  of  a  table  given  on  page  248  of  Kohl- 
rausch's  Kleiner  Leitfaden  der  Praktischen  Physik. 
APPARATUS  No.  3, 

In  order  to  have  a  second  piece  of  apparatus  available  for 
experiments  with  ether,  so  that  two  experiments  could  be  made 
at  the  same  tmie,  a  third  piece  of  apparatus  was  constructed. 
The  mercury  tube  of  this  apparatus  was  made  a  little  smaller 
than  that  of  apparatus  No.  2,  as  some  difficulty  had  been  met 
in  the  mercury  column  failing  to  hold  together  properly. 
The  capillary  of  this  apparatus  was  also  elliptical  and  slightly 
conical.  The  average  semi -major  axis  as  determined  by  the 
microscope  and  mercury  was  .006176  cm.  while  the  minor  axis 
was  .002837.  The  length  of  the  capillary  was  73.25  cm.  at  first, 
but  was  reduced  to  70  cm.  when  the  apparatus  was  repaired 
after  a  breakage. 

The  distance  between  the  marks  on  the  mercury  tube  was 
38.45  cm.  and  the  volume  of  gas  transpired  was  .742  c.c.  since 
10.0634  grams  of  mercury  occupied  the  distance  between  the 
marks  at  16°  C.  The  average  diameter  of  the  mercury  tube 
where  the  mercury  traversed  it  during  an  experiment  was 
.158112  cm. 

A  table  of  the  results  obtained  with  this  apparatus  for  air 


46 

is  given  on  page  71  underneath  those  for  air  with  apparatus 
No.  2.  It  will  be  noticed  that  the  two  pieces  of  apparatus 
agree  very  closely  in  results. 

TABLE  OF  INTERNAL  FRICTION  OF  ETHER  GASES. 

APPARATUS  No.  3. 
Ht.  of  Press  of 


Length 
Capillary 

Kind 
of  Gas 

Temp,    driving 
of  Gas  Col. 
atO°C 

Barom. 
at  0°  C 

entering   Time  in 
Gas       seconds     B  X  1  07 

Average 

70. 

Di-meth 

100.3 

18.75 

77.24 

95. 

03 

315.6 

1191. 

8 

70. 

yl  Ether 

100.3 

18.74 

77.32 

95.10 

314.8 

1188.6 

1190.2-j-l. 

70. 

Ethyl 

100. 

3 

18.74 

76.98 

94. 

67 

303.6 

1145. 

6 

70. 

Alcohol 

100. 

3 

18.76 

76.99 

94 

.70 

308.4 

1164 

.4 

1155.    -J-6.3 

73.25 

Di-ethyl 

100. 

1 

18.78 

76.43 

94 

.26 

287.4. 

1041 

.8 

73.25 

Ether 

100. 

18.60 

76.15 

93 

.81 

287.4 

1023.2 

73.25 

100. 

18.79 

76.03 

93. 

88 

270.4 

978. 

4 

1001.    -[-8.15 

70. 

99. 

9 

18.76 

75.80 

93.61 

254.8 

963 

.7 

70. 

100. 

1 

18.56 

76.46 

94 

.07 

266.5 

998 

.1 

73.25 

Ethyl- 

99. 

9 

18.76 

75.79 

93 

61 

253  .  2 

914 

.8 

73.25 

Propyl 

99. 

9 

18.26 

75.74 

93. 

06 

267.2 

939.3 

73.25 

Ether 

100. 

18.05 

76.39 

93. 

50 

262.4 

921. 

7 

918.4-J-4.2 

73.25 

100. 

3 

18.28 

76.83 

94 

.16 

256.6 

902 

.7 

73.25 

100 

4 

18.05 

77.02 

94 

.14 

263.2 

913 

.7 

70. 

Ethyl- 

100. 

2 

17.74 

76.46 

93, 

26 

267.7 

946 

,4 

70. 

Isopro- 

100. 

2 

18.76 

76.41 

94 

.22 

255.4 

966 

.4 

960.7-J-5. 

70. 

pyl  Ether  100  .  2 

18.56 

76.34 

93 

.95 

258.8 

969 

.3 

70. 

Di-Pro- 

100. 

18.75 

75.95 

93.73 

222  2 

839. 

5 

70. 

pyl 

100. 

18.76 

76.02 

93 

.74 

222.0 

839 

.3 

837.  5  -J-  1.26 

70. 

Ether 

100. 

3 

18.75 

77.14 

94 

93 

220.8 

832 

.7 

70. 

Isopro- 

100. 

1 

18.73 

76.19 

93. 

97 

230.4 

870.2 

873.4-1-2.2 

70. 

pyl-pro- 

100. 

1 

18.76 

76.22 

94.03 

231.6 

876 

.6 

pyl  Ether 

70. 

Di-iso- 

100. 

3 

18.74 

77.14 

94 

.92 

230.2 

869 

.2 

70. 

propyl 

100. 

3 

18.74 

77.14 

94 

.92 

230.8 

871.5 

70. 

Ether 

100. 

3 

18.76 

77.09 

94 

.89 

239.0 

902 

.8 

70. 

100. 

3 

18.74 

77.13 

94. 

91 

236.0 

890.7 

894.  3  -j-  5.  95 

70. 

100. 

1 

18.73 

76.23 

94 

.01 

241.0 

910 

.5 

70. 

100. 

1 

18.75 

76.23 

94, 

,03 

243.6 

921 

2 

The  results  obtained  for  ethers  with  this  apparatus  are 
given  in  the  above  table.  The  first  point  worthy  of  note  is 
that  ethyl  alcohol  which  is  metameric  with  di-methyl  ether 
has  about  3%  lower  coefficient  of  friction,  showing  it  to  have 
the  larger  molecule, 

The  values  for  di-ethyl  ether,  and  ethyl-propyl  ether  agree 
well  with  those  given  on  page  43  by  apparatus  No.  2. 


47 


Ethyl-isopropyl  ether  has  a  friction  4.6%  higher  than  that 
of  ethyl  propyl  ether,  hence  has  a  smaller  molecule. 

Ethyl-propyl  ether 


Ethyl-isopropyl  ether  H3C  —  C  —  O  —  C<3 

H2          H    CHa 

Isopropyl  -propyl  ether  has  a  friction  4.3%  higher  than  that 
of  di-propyl  ether,  while  di-isopropyl  ether  has  a  friction  6.8% 
higher  than  di-propyl.  This  is  in  accordance  with  our  view 
of  the  concatenation  of  the  atoms  in  these  molecules.  Di- 
propyl  ether  is  a  straight,.  chain  compound  whereas  iso-propyl 
propyl  has  two  branches  and  di-isopropyl  has  four  branches. 

Di-propyl  ether 

Isopropyl-propyl  ether  u3n>C  — 

h3<-     H  H2  H2  H3 

Di-isopropyl  ether         **,C  >  C—  O—  C  <  £§» 

3       H  H          3 

As  already  noticed  the  difference  between  methyl-propyl 
and  methyl  -isopropyl  is  5%  due  to  2  branches;  the  difference 
between  ethyl-propyl  and  ethyl-isopropyl  due  to  two  branches 
is  4.6%;  the  difference  between  di-propyl  and  isopropyl-propyl 
is  4  .  3  %  due  to  the  same  number  of  branches.  The  diminishing 
effect  of  the  2  CH3  branches  is  probably  due  to  the  fact  that 
the  molecules  of  the  substances  lower  on  the  lists  are  larger, 
hence  the  branches  have  a  smaller  relative  effect.  It  would  seem 
that  di-isopropyl  ether  ought  to  be  8.6%  higher  than  di- 
propyl  whereas  it  is  only  6  .  8%  .  This  discrepancy  I  am  unable 
to  explain. 

Di-methyl  ether,  being  a  gas  at  ordinary  temperatures, 
had  to  be  handled  differently  from  the  other  ethers,  all  of  which 
were  put  into  the  apparatus  as  liquids.  The  low  boiling  ethers, 
methyl-ethyl,  and  methyl-isopropyl  were  experimented  upon 
in  a  room  whose  temperature  was  at  or  very  near  0°  C. 

The  di  -methyl  ether  gas  was  kept  in  sealed  glass  bulbs  of  250  c.c 
capacity.  The  lower  end  of  a  bulb  was  connected  by  rubber 
tubing  to  a  reservoir  of  mercury  while  the  upper  end  was  connected 
by  a  rubber  tube  to  the  funnel  at  the  top  of  the  apparatus. 


48 

After  the  aspirator  had  created  a  partial  vacuum  in  the 
apparatus  by  sucking  the  air  out  through  the  capillary,  the 
tips  of  the  bulb  were  broken  off  inside  the  rubber  tubes 
and  mercury  allowed  to  flow  into  the  bulb  gradually  from 
the  bottom,  driving  the  ether  gas  before  it  into  the  ap- 
paratus, the  action  being  assisted  all  the  while  by  the  suction  of 
the  aspirator.  As  250  c.c.  was  many  times  the  cubic  capacity 
of  the  apparatus,  by  the  time  the  mercury  had  entirely  filled  the 
ether  bulb,  it  was  judged  the  apparatus  would  be  filled  with 
pure  ether,  unmixed  with  any  air.  This  method  of  getting  the 
ether  gas  into  the  apparatus  proved  a  failure  several  times  for 
various  reasons,  but  the  very  close  agreement  of  the  two  read- 
ings which  were  at  length  obtained  points  to  their  correctness. 

MOLECULAR  VOLUMES. 

L.  Meyer1  gives  the  following  formula  for  calculating  approxi- 
mate relative  molecular  volumes,  derived  from  the  molecular 
volume  of  5  02  as  determined  by  Andreef  at  -8°  C 


V  ==  3.10"6  j    M  (1  +at)    I*  j  — 


in  which  M  =  molecular  weight,  t  =  temperature  centigrade  T?  = 
coefficient  of  friction. 

The  values  obtained  for  the  substances  under  investigation 
are  given  on  the  next  page.  According  to  Kopp1  the  molecular 
volume  of  a  liquid  composed  of  carbon,  hydrogen  and  oxygen 
can  be  found  by  substituting  in  its  formula  11  for  each  atom  of 
carbon,  5.5  for  each  hydrogen  and  7.8  for  each  oxygen.  The 
values  thus  calculated  are  given  in  the  last  column.  The  agree- 
ment between  the  molecular  volumes  calculated  in  these  two 
ways  is  quite  good,  much  better  than  that  obtained  by  L.  Meyer 
and  Schumann  (p.  21). 

As  both  apparatus  No.  2  and  No.  3  gave  values  for  air  about 
5%  higher  than  those  given  by  Landolt  and  Boernstein  as  prob- 
ably the  most  correct  values,  it  is  probable  that  the  values  of 
>?  for  the  ethers  are  correspondingly  too  high.  Accordingly  in 
the  second  table  of  molecular  volumes  the  values  of  IQ  were 
reduced  by  multiplying  by  2113/2244,  thus  calibrating  with 
air,  and  the  molecular  volumes  recalculated.  A  still  better 


1.   Wied.  Ann.,  1881,  Bd.  13,  p.   17. 

1.  Ann.  Chem.  Phar.,  1855,  Bd.  96,  pp.  1,  153,  303. 


49 


agreement  with  Kopp's  values  is  shown,  the  disagreement  being 
greatest  when  the  boiling  point  of  the  ether  approaches  the  tem- 
perature at  which  i)  was  determined. 


FIRST  TABLE  OF  MOLECULAR  VOLUMES. 


Material 

Boiling 
Point 
Formula      Cent.        r/XlO7 

Molec.  V.  Molec.  V. 
Found    Calculated 
according 

to 

Kopp 

Di-methyl  ether 

C2H 

6° 

-23.65 

1190.2 

51.62 

I 

62 

.8 

Ethyl  alcohol 

CTT 
2 

6° 

78. 

1155. 

0 

55.25 

s 

Methyl-ethyl  ether 

CTT 
3 

8° 

10.  -13. 

1092. 

3 

71.65 

84 

.8 

Di-ethyl  ether 

CTT 
4 

10° 

34.6-35. 

1000. 

95.73 

) 

Methyl-propyl  ether 

CTT 
jXi 

10° 

40-45. 

1004. 

3 

95.16 

106 

.S 

Methyl-isopropyl  ether 

CTT 
4 

10° 

7. 

1054. 

s 

88.37 

) 

Ethyl-propyl  ether 

C5H 

12° 

66.  -68- 

910. 

8 

124.19 

I 

1  9S 

Ethyl-isopropyl  ether 

CTT 
-IT. 

12° 

54.  -57. 

960. 

7 

115.77 

» 

LAO 

Di-propyl  ether 

C6H 

84.5-86.5 

837. 

5 

158.90 

Isopropyl-propyl-ether 

CTT 
piJ. 

uO 

76.5-77. 

873. 

4 

148.86 

[ 

150 

.8 

Di-isopropyl  ether 

CTT 

8 

14° 

68.5-69. 

894. 

3 

144.06 

) 

SECOND  TABLE  OF  MOLECULAR  VOLUMES. 


Material 

Boiling 
Point  in 
Formula  deg.  Cent.  7?X107 

Molec.  V. 
Molec.  V.  Calculated 
Found    according 
to  Kopp 

Di-methyl  ether 
Ethyl  alcohol 
Methyl-ethyl  ether 

cX° 

C3H8O 

-23.65  1121 
78.        1088 
10.  -13.      1029 

.0 
.0 
.0 

56. 
59. 

81. 

46) 
05) 
12 

62, 

84. 

,8 
8 

Di-ethyl  ether 

C 

4H10O 

36. 

,6-35.     942 

.0 

104. 

71  ? 

Methyl-propyl  ether 

C 

4H10° 

40. 

-45. 

946 

.0 

104. 

20  [ 

106. 

8 

Methyl-isopropyl  ether 

C 

*H100 

7. 

993 

.6 

96. 

66  ) 

Ethyl-propyl  ether 

C 

5H12° 

66. 

-68. 

863 

.6 

135. 

84  | 

1  28 

Ethyl-isopropyl  ether 

C 

5H12° 

54, 

,-57. 

905 

.1 

126. 

61  f 

1  <6o  < 

Di-propyl  ether 

C 

6H140 

84 

,5-86. 

5  788 

.9 

173. 

08^ 

Isopropyl-propyl-ether 

C 

6H14° 

76. 

5-77. 

822 

.7 

163. 

20  > 

150. 

8 

Di-isopropyl  ether 

C 

6H140 

68. 

5-69. 

842 

.4 

157. 

5lJ 

In  this  second  table  r)  has  been  reduced  by  calibration  with  air. 


MOLECULAR  MEAN  SPEEDS,  FREE  PATHS  AND  COLLISION 

FREQUENCIES. 

In  Meyer's  Kinetic  Theory  of    Gases  on  page  219  is  given  the 
formula  y  =0  .  30967  p  LSI  whence 


T      . 


0.30967  ptt 

in  which  i)  =  coefficient  of  internal  friction  of  the  gas 
p  =  density  of  the  gas 
Q  =mean  molecular  velocity 
L  =mean  molecular  free  path. 


On  page  55  of  the  same  work  we  find 

^f- 

where  p  —  pressure  in  absolute  measure. 
On  page  195  we  see  that  the 


p  =  —npQ2    whence 

o 


Q 


Collision  Frequency  =  -=- 


The  values  given  in  the  table  on  this  page  were  calcu- 
lated according  to  the  above  formulae,  at  a  pressure  of  76  cm. 
of  mercury  and  a  temperature  of  100°C.,  the  density  being  cal- 
culated from  the  formula 

(formula   weight)    (0.000089) 

It  will  be  noticed  that  the  values  are  all  of  the  proper  order  of 
magnitude,  but  do  not  rise  and  fall  in  a  periodic  way  while  the 

FIRST  TABLE  OF  MEAN  MOLECULAR  SPEEDS,  FREE  PATHS  AND  COLLISION 

FREQUENCIES. 


Mean 

Molecular 

Collision 

Molecular 

Molecular 

Free  PathFrequency 

Material 

We 

ight 

7?X10 

7 

speed  cm. 

cm.XlO10 

xio-6 

per  Sec. 

Di-methyl  ether 

46 

.048 

1190. 

2 

41470 

61787 

6711. 

7 

Ethyl  alcohol 

46 

.048 

1155. 

0 

41470 

59970 

6915. 

0 

Methyl-ethyl  ether 

60 

.064 

1092. 

3 

36310 

49660 

7312. 

0 

Di-ethyl  ether 

74 

.08 

1000. 

0 

32695 

40895 

7994. 

9 

Methyl-propyl  ether 

74 

.08 

1004. 

3 

32695 

41111 

7952. 

7 

Methyl-isopropyl  ether 

74 

.08 

1054. 

8 

32695 

43180 

7572. 

0 

Ethyl-propyl  ether 

88.096 

916.8 

29982 

34415 

8711. 

8 

Ethyl-isopropyl  ether 

88 

.096 

960. 

7 

29982 

36063 

8315. 

5 

Di-propyl-propyl  ether 

102 

.112 

837. 

5 

27848 

29201 

9536. 

0 

Isopropyl-propyl  ether 

102 

.112 

873. 

4 

27848 

30454 

9144. 

G 

Di-isopropyl  ether 

102 

.112 

894. 

3 

27848 

31182 

8931. 

0 

51 


SECOND  TABLE  OF  MEAN  MOLECULAR  SPEEDS,  FREE  PATHS  AND  COLLI- 
SION FREQUENCIES. 


Di-methyl  ether 

46. 

.048 

1121 

.0 

41470 

58204 

7125. 

0 

Ethyl  alcohol 

46 

.048 

1088 

.0 

41470 

56490 

7341. 

0 

Methyl-ethyl  ether 

60. 

064 

1029 

.0 

36310 

46782 

7762. 

0 

Di-ethyl  ether 

74 

.08 

942 

.0 

32695 

38570 

8477. 

0 

Methyl-propyl  ether 

74, 

,08 

946 

.0 

32695 

38725 

8639. 

5 

Methyl-isopropyl  ether 

74. 

08 

993 

.6 

32695 

40682 

8036. 

7 

Ethyl-propyl  ether 

88. 

096 

863 

.6 

29982 

32418 

9248. 

4 

Ethyl-isopropyl  ether 

88. 

096 

905 

.1 

29982 

33976 

8824. 

3 

Di-propyl  propyl  ether 

102. 

112 

788 

.9 

27848 

27507 

10124. 

9 

Isopropyl-propyl  ether 

102. 

112 

822 

.7 

27848 

28685 

9708. 

9 

Di-isopropyl  ether 

102. 

112 

842 

.4 

27848 

29372 

9481. 

1 

In  this  second  table  y  has  been  reduced  by  calibration  with  air. 


molecular  weight  increases,  as  pointed  out  for  other  substances 
by  Meyer  on  page  196  of  his  Kinetic  Theory  of  Gases,  probably 
because  these  ethers  are  all  so  closely  related. 

COMPARISON  WITH  THE  RESULTS  OF  OTHERS. 

The  results  of  my  work  cannot  be  compared  satisfactorily 
and  exactly  with  that  of  others,  because  I  have  determined  the 
friction  at  a  temperature  higher  than  most  other  observers,  and 
the  law  of  its  increase  with  the  temperature  is  not  accurately 
known  in  each  case.  Furthermore,  air,  di-ethyl  ether,  dimethyl 
ether  and  ethyl  alcohol  are  the  only  substances  which  I  have 
employed  that  others  have  investigated. 

Disregarding  the  results  with  apparatus  No.  1  as  being  only 
preliminary,  we  see  that  the  values  of  tj  for  air  for  the  other 
two  pieces  of  apparatus  agree  closely  and  are  consistent,  though 
both  are  about  5%  higher  than  the  values  which  are  most 
probably  correct,  according  to  Landolt  and  Boernstein's  tables. 
That  the  relative  values  of  y  at  the  different  temperatures  are 
correct  is  shown  by  calculating  y  at  0°  C.  by  Sutherland's  for- 
mula (see  page  30)  and  then  calculating  from  tha+  y  at  100°  C. 
Using  the  value  of  the  cohesion  constant  C=119.4  as  deter- 
mined by  Breitenbach  we  get  for  apparatus  No.  2,  >?0  =  .0001793 
and  T?IOO=  .00022367  the  latter  value  agreeing  with  .0002242 
observed  within  1/4%.  For  apparatus  No.  3  we  get  in  like 
manner  >?0  =  .  0001795  and  ^100=  .0002239,  the  latter  value 
agreeing  with  .0002246  observed  within  \%.  This  agreement 
is  within  the  limits  of  error  of  the  experiments. 


52 

Puluj's  formula  for  the  coefficient  of  friction  of  di-ethyl 
ether  vapor,  y  =0.0000689  (1  +.0041575  t)  *94  gives j  =0.0000935 
at  100°  C.  which  agrees  with  my  corrected  value  of  0.0000942 
within  }%.  This  is  good  agreement  bearing  in  mind  the  fact 
that  Puluj's  formula  was  determined  from  experiments  over  a 
small  range  of  temperature,  viz.:  from  0°  to  37°  C. 

The  coefficient  of  friction  of  di-methyl  ether  is  given  on 
page  192  of  Meyer's  Kinetic  Theory  of  Gases  as  0.000092  at  0°  C. 
No  one  has  determined  the  law  of  its  increase  with  the  tem- 
perature. My  corrected  value  is  .0001121  at  100°  C.  which 
seems  reasonable. 

Steudel  (see  page  21)  gives  0.000142  as  the  coefficient  of 
friction  of  ethyl  alcohol  at  78.4°  C.,  its  boiling  point.  My  cor- 
rected value  of  .0001088  at  100°  C.  does  not  agree  well  with 
this.  Because  his  value  was  determined  at  the  boiling  point 
I  think  it  is  open  to  question.  My  value  for  ethyl  alcohol 
agrees  much  better  with  Puluj's1  results  which  are  0.0000827 
at  0°  C.  and  0.0000885  at  16.8°  C.  He  assumes  y  is  proportional 
to  the  absolute  temperature  which  gives  y  =  .0001130  at  100°  C., 
which  is  nearly  4%  higher  than  my  value  of  .0001088.  I  think 
the  assumption  that  ^  increases  exactly  as  the  absolute  tem- 
perature is  not  strictly  correct. 

Concerning  the  differences  which  I  have  found  between  the 
normal  propyl  and  the  iso-propyl  ethers  I  would  point  out  that 
Lothar  Meyer,  Schumann  and  Steudel  (see  pages  20  to  23) 
found  similar  differences  between  many,  though  not  all,  of  the 
normal  propyl  and  iso-propyl  compounds  which  they  examined. 
They  also  found  that  normal  butyl,  isobutyl  and  tertiary 
butyl  compounds  showed  still  more  regular  differences.  The 
weight  of  evidence  gives  a  larger  molecule  to  the  primary,  a 
smaller  to  the  secondary  and  the  smallest  to  the  tertiary  com- 
pound. My  values  of  the  molecular  speeds,  free  paths  and 
collision  frequencies  being  for  100°  C.  of  course  do  not  agree 
with  those  calculated  for  0°  C.  by  others. 

SUMMARY  OF  RESULTS. 

1.  A  new  and  simple  apparatus  for  determining  the  internal 
friction  of  gases  and  vapors  has  been  developed. 

2.  The  coefficients  of  internal  friction  of  the  following  eight 
ether  gases  which  have  hitherto  not  been  experimented  with, 
have  been  determined  with  considerable  accuracy  as  follows: 


53 

Methyl-ethyl  ether 0.0001029    at  100°  C. 

Methyl-propyl  ether 0.0000946     " 

Methyl-isopropyl  ether .0.00009936  " 

Ethyl-propyl  ether 0.00008636  " 

Ethyl-isopropyl  ether. 0.00009051   " 

Di-propyl  ether 0.00007889  " 

Iso-propyl-propyl  ether .0.00008227  " 

Di-isopropyl 0.00008424  " 

3.  The  molecular  volumes  calculated  from  the  friction  have 
T}een  shown  to  agree  fairly  well  with  those  obtained  by  Kopp's 
rule. 

4.  A  marked  and  unmistakable  difference  between  the  nor- 
mal propyl  and  isopropyl  ethers  has  been  found,  proving  that 
the  difference  in  the  molecular  structure  of  these  ethers  has  a 
very  noticeable  effect  upon  their  internal  friction,  and  there- 
fore upon  the  size  of  their  molecules,  the  molecules  having 
the  most  numerous  branches  being  smaller  than  those  with 
fewer  or  no  branches.     The  object  of  this  research  has  therefore 
been  accomplished. 

In  conclusion  I  wish  to  express  my  most  sincere  and  heartfelt 
thanks  to  Professor  Loeb  for  suggesting,  to  me  both  the  subject 
of  this  research  and  the  form  of  apparatus,  and  for  his 
kind  interest  and  help  throughout  the  course  of  the  investiga- 
tion. In  addition  I  wish  to  thank  him  heartily  for  his  very 
great  courtesy  in  supplying  me  with  the  necessary  ethers. 

I  also  wish  to  express  my  thanks  to  Professors  Charles  Basker- 
ville  and  Charles  A.  Doremus  of  the  College  of  the  City  of  New 
York,  for  their  kindness  in  permitting  me  to  perform  most  of 
the  experiments  in  the  chemical  laboratory  of  that  institution, 
in  order  to  save  time  in  going  to  and  from  the  laboratory  of 
New  York  University. 

FREDERICK  M.  PEDERSEN. 

New  York  University,  April,  1905. 


BIBLIOGRAPHY. 
BAILY. 

Phil.  Trans.,  1832,  p.  399. 
BARUS. 

Am.  Journ.  Sci.,  1888  (3),  Vol.  35,  p.  407. 

Bull.  U.  S.  Geol.  Survey,  No.  54,  Washington,  1889,  p.  39. 

Wied.  Ann.,  1889,  Bd.  36,  p.  358. 

Phil.  Mag.,  1890,  Vol.  29,  p.  337. 
BERNOULLI,  JOHANN. 

Opera  omnia.     Lausannae  et  Genevae,  1742,  Tomus  III.     Nouvelles 

pense"es  sur  le  systeme  de  M.  Descartes  XIX.-XXIII. 
BESSEL. 

Abh.  d.  Berl.  Akad.  Math.  Klasse,  1826,  p.  1. 
BESTELMEYER. 

Inaug.  Diss.  Munich,  1902. 

Ann.  d.  Phys.,  1904,  Vol.  13,  p.  944. 

BOLTZMANN. 

Wien.  Ber.  Math.  Naturw.,  1868,  Bd.  58  (2),  p.  517. 

1872,     "    66  (2),  p.  324. 

1880,  "    81  (2),  p.  117. 

1881,  "    84  (2),  pp.  40,  1230. 

1887,  "    95  (2),  p.  153. 

1888,  "    96  (2),  p.  891. 
Wied.  Ann.,  1897,  Bd.  60,  p.  399. 

BRAUN  AND  KURZ. 

Carls  Rep.,  1882,  Vol.  18,  pp.  569,  665,  697. 

1883,  Vol.  19,  pp.  343,  605. 
BREITENBACH. 

Wied.  Ann.,  1899,  Bd.  67,  p.  803. 

Ann.  d.  Phys.,  1901,  Bd.  5,  p.  166. 
CAUCHY. 

Exerc.  de  Mathe"m.,  1828,  p.  183. 
CHALLIS. 

Phil.  Mag.,  1833,  p.  185. 
CHEZY. 

Me*m.  Manuscript  de  1'Ecole  des  Fonts  et  Chausse*e,  1775. 
CLAUSIUS. 

Pogg.  Ann.,  1858,  Bd.  105,  p.  239. 
CLEBSH. 

Crelle's  Journ.  fur  Math.,  Bd.  52,  1856,  p.  119. 
COUETTE. 

Compt.  Rend.,  1888,  Vol.  107,  p.  388. 

Journ.  de  Phys.,  1890  (2),  Vol.  9,  p.  414. 

Ann.  de  Chim.  et  de  Phys.,  1890  (6),  Vol.  21,  p.  433. 

54 


55 


COULOMB. 

Me'm.  de  1'Institut  National.     Year  9  (1801),  Tome  III.,  p.  246. 
COUPLET. 

Me'm.  de  1' Academic,  1732. 
CROOKES. 

Phil.  Trans.,  1881,  Vol.  172,  p.  387. 

D'ALEMBERT. 

Trait^  de  I'^quilibre  et  du  mouvement  des  fluides,   nouvelle  e"dit. 
Paris,  1770. 

D'ARCY. 

Me'm.  de  Divers  Savans,  1858,  Vol.  15,  p.  141. 
DE  KEEN. 

Bull,  de  1'Acad.  de  Belgiques,  1888  (3),  Vol.  16,  p.  195. 
Du  BUAT. 

Principes  d'Hydraulique,  1786. 
EULER. 

Tentamen  theoriae  de  frictione  fluidorum.    Novi  commentarii  Ptero- 
politani,  tomus  VI.,  1756,  et  57  Pag.  338.     Die  Gesteze  des  Gleich- 
gewichtes  und  der  Bewegung  fliissiger  Korper.     Translated  from 
the  Latin  by  W.  Brandes.     Leipzig,   1806. 
EYTELWEIN. 

Abh.  d.  Berl.  Akad.,  1814  and  1815. 
GERSTNER. 

Neu.  Abh.  der  kon.  Bohmischen  Gesell.  der  Wiss.,  Bd.  3,  Prag  1798. 

Gilbert's  Annalen,  Bd.  5,  1900,  p.  160. 

GlRARD. 

Me'm.  de  1'Institut.,  Classe  Sc.  Math.,  1813-15,  p.  248;  1816,  p.    187 

GlRAULT. 

Me'm.  de  1'Acad.  de  Caen,  1860. 
GIULIO. 

Memorie  di  Torino,  Ser.  2,  tomo  13,   1853. 
GRAHAM. 

Phil.  Trans.,  1846,  Vol.  136,  pp.  573,  622. 
1849,  Vol.  139,  p.  349. 

Pogg.  Ann.,  1866,  Bd.  127,  pp.  279,  365. 
GREEN. 

Transactions  of  the  Royal  Society  of  Edinburgh,  1836,  Vol.  13,  p.  54. 
GRONAU. 

Uber  die  Bewegung  schwingender  Korper  im  widerstehenden  Mittel. 
Danzig,  1850. 

Program  der  Johannesschule. 
GROSSMAN. 

Inaug.  Diss.  Breslau,  1880. 

Wied.  Ann.,  1882,  Bd.  16,  p.  619. 
GROTRIAN. 

Pogg.  Ann.,  1876,  Vol.  157,  pp.  130,  237. 
41       1877,  Vol.  160,  p.  238. 

Wied.  Ann.,  1879,  Vol.  8,  p.  529. 

GUTHRIE. 

Phil.  Mag.,  1878  (5),  Vol.  5,  p.  433. 


56 

HAGEN. 

Pogg.  Ann.,  1839,  Bd.  46,  p.  423. 

Abth.  d.  Berl.  Akad.,  1854,  p.  17. 
HAGENBACH. 

Pogg.  Ann.,  1860,  Bd.  109,  pp.  385,  401. 
HELMHOLTZ  AND  PIETROWSKI. 

Wien.  Ber.  Mathem.  Naturw.,  1860,  Vol.  40,  p.  607. 
HOFFMANN. 

Wied.  Ann.,  1884,  Bd.  21,  p.  470. 
HOLM  AN. 

Proc.  Am.  Acad.  Boston,  1877,  Vol.  12,  p.  41. 
1886,  Vol.  21,  p.  1. 

Phil.  Mag.,  1877  (5),  Vol.  3,  p.  81;  1886,  Vol.  21,  p.  199. 

HOUDAILLE. 

Fortschr.  d.  Phys.,  1896,  52  Jahr.,  I.,  p.  442. 
JACOBSON,  H. 

Archiv.  fur  Anatomic  und  Physiologic  von  Reichert  und  Du  Dois 

1860  and  1861. 
JAEGER. 

Wien.  Ber.  Mathm.  Naturw.,  1899  (2),  Vol.  108,  p.  447. 

1900  (2),  Vol.  109,  p.  74. 
JOB. 

Bull.  Soc.  Franc.  Phys.,  1901,  Vol.  157,  p.  2. 
KLEMENCIC. 

Carls.  Rep.,  1881,  Bd.  17,  p.  144. 

Wien.  Ber.,  1881  (2),  Bd.  84,  p.  146. 

Beiblatter,  1882,  Bd.  6,  p.  66. 

KlRCHHOFF. 

Mechanik.,  1877,  4  Aufl.,  26  Vorl.,  p.  383. 
KOCH. 

Wied.  Ann.,  1883,  Bd.  19,  p.  857. 

KOENIG. 

Wied.  Ann.,  1885,  Bd.  25,  p.  618;  1887,  Bd.  32,  p.  193. 

Sitz.  d.  Munch.  Akad.,  1887,  Bd.  17,  p.  343. 
KUNDT  AND  WARBURG. 

Monatsber.  d.  Berl.  Akad.,  1875,  p.  160. 

Pogg.  Ann.,   1875,  Vol.  155,  pp.  337,  525. 

Phil.  Mag.,  1875  (4),  Vol.  50,  p.  53. 
KLEINT. 

Inaug.  Diss.  Halle,  1904. 
LAMPE. 

Programm  des  Stadtischen  Gym.  zu  Danzig,   1866. 
LAMPEL. 

Wien.  Ber.  Mathem.  Naturw.,  1886,  Bd.  (2),  p.  291. 
LANG. 

Wien.  Ber.  Math.  Naturw.,  1871,  Vol.  63  (2),  p.  604. 

1872,  Vol.  64  (2),  p.  487. 

Pogg.  Ann.,  1872,  Vol.  145.  p.  290. 
1873,  Vol.  148,  p.  550. 


57 


LUDWIG  AND  STEFAN. 

Sftzber.  Wien.  Akad.,  1858,  Vol.  32,  p.  25. 
MARIAN. 

Histoire  de  1'Acad.  de  Paris,  1735,  p.  166. 
MARGULES. 

Wien.  Ber.  Mathem.  Naturw.,  1881,  Vol.  83  (2),  p.  588. 
MARKOWSKI. 

Inaug.  Diss.  Halle,  1903. 

Ann.  d.  Phys.,  1904,  Vol.  14,  p.  742. 
MAXWELL. 

Phil.  Mag.,  1860  (4),  Vol.  19,  p.  31. 

1868  (4),  Vol.  25,  p.  209  and  211. 

Phil.  Trans.,  1866,  Vol.  156  (1),  p.  249. 

Collected  Papers,  Vol.  II. 

Proc.  Royal  Society,  1866,  Vol.  15,  p.  14. 
MATHIEU. 

Compt.  Rend.,  1863,  Tome  57,  p.  320. 
MEYER,  LOTHAR. 

Ann.  d.  Chem.  u.  Phar.,  1867,  Suppl.  Bd.  5,  p.  129. 

Wied.  Ann.,  1879,  Bd.  7,  p.  497. 

1882,  Bd.  16,  p.  394. 
MEYER,  L.  AND  SCHUMANN. 

Wied.  Ann.,  1881,  Bd.  13,  p.  1. 
MEYER,  O.  E. 

Dissertation:  de  mutua  duorum  fluidorum  frictione. 

Regimonti's  Prussorum,  1860. 

Crelle's  Journal  fur  Mathem.,  1861,  Bd.  59,  p.  229. 

Pogg.  Ann.,  1861,  Bd.  113,  p.  55;  1865,  Bd.  125,  pp.  177,  401,  564; 
1866,  Bd.  127,  pp.  253,  353;  1871,  Bd.  142,  p.  513;  1871,  Bd.  143, 
p.  14;  1873,  Bd.  148,  p.  203. 

Wied.  Ann.,  1887,  Bd.  32,  p.  642;  1891,  Bd.  43,  p.  1. 

Sitz.  d.  Munchen  Akad.,  1887,  Bd.  17,  p.  343. 
MEYER,  O.  E.  AND  SPRINGMVHL. 

Pogg.  Ann.,  1873,  Vol.  148,  p.  526. 
MORITZ. 

Pogg.  Ann.,  Bd.  70,  p.  74. 
MUTZEL. 

Wied.  Annv  1891,  Bd.  43,  p.  15. 
NAUMANN. 

Ann.  d.  Chem.  u.  Pharm.,  1867,  Suppl.  Bd.  5,  p.  252. 
NAVIER. 

Me'm.  de  1'Acad.  des  Sciences,  1823,  Tome  6,  p.  389. 

1830,  Tome  9,  p.  311. 
NOYES  AND  GOODWIN. 

Physical  Review,  1896,  Vol.  4. 

Zeitsch.  Physik.  Chem.,  1896,  Vol.  21,  p.  671. 
NEUMANN. 

Einl.  in  d.  theor.  Physik.,  1883,  p.  246. 
NEWTON. 

Philosophiae  naturalis  principia  mathematica,  1687,  Lib.  II.,  Sect.  IX. 


58 

OBERMAYER. 

Wien.  Ber.  Mathem.  Naturw.,  1875,  Vol.  71  (2),  p.  281. 

1876,  Vol.  73  (2),  p.  433. 

Carls.  Rep.,  1876  (2),  Vol.  12,  pp.  13,  456. 
"      1877,  .Vol.  13,  p.  130. 

Phil.  Mag.,  1886,  Vol.  21. 
ORTLOFF. 

Inaug.  Diss.  Jena,  1895. 
PEROT  AND  FABRY. 

Compt.  Rend.,  1897,  Vol.  124,  p.  28. 

Ann.  de  Chim.  et  de  Phys.,  1898  (7),  Vol.  13,  p.  275. 

POISEUILLE. 

Soc.  Philomath,  1838,  p.  77. 

Compt.  rend.,  1840,  Vol.  11,  pp.  961,  1041. 

1841,  Vol.  12,  p.  112. 

1842,  Vol.  15,  p.  1167. 

Ann.  de  Chim.  et  de  Phys.,  1843  (3),  Vol.  7,  p.  50. 

Me*m.  de  Savans  Strangers.  1846,  Vol.  9,  p.  433. 
POISSON. 

Journal  de  1'Ecole  Polytech.,  1831,  20  me  cahier,  tome  13,  p.  139. 

Connaissance  des  Terns,   1834.     Appendix. 

M(§m.  de  1'Acad.,  Tome  2,  1832,  p.  521. 
PLAN A. 

Me'm.  de  1'Acad.  di  Torino,  T  37,  1835. 
PRONY. 

Recherches  physico-mathe'matiques  sur  la  the*orie  des  eaux  courantes. 
Paris,  1804. 

PULUJ. 

Wien.  Ber.  Mathem.  Naturw.,  1874,  Bd.  69  (2),  p.  287. 

1874,  Bd.  70  (2),  243. 
1876,  Bd.  73  (2),  589. 
1878,  Bd.  78  (2),  p.  279. 
Carls.  Rep.,  1878,  Bd.  14,  p.  573. 

"       1879,  Bd.  15,  pp.  427,  578,  633. 
Phil.  Mag.,  1878  (5),  Vol.  6,  p.  157. 
RAYLEIGH. 

Proc.  Roy.  Soc.,  1900,  Vol.  66,  p.  68;  Vol.  67,  p.  137 
RELLSTAB. 

Inaug,  Diss.  Bonn.,  1868. 
REYNOLDS,  F.  G. 

Phys.  Rev.,  1904,  Vol.  18,  p.  419;  Vol.  19,  p.  37. 
REYNOLDS,  O. 

Proc.  Roy.  Inst.  Grt.  Brit.,   1884,  Vol.   11,  p.  44. 
Beiblatter,  1886,  Bd.  10,  p.  217. 
SABINE. 

Phil.  Trans.,   1829,  p.  207  and  331;   1831,  p.  470. 

SCHNEEBELI. 

Arch,  de  Geneve,   1885,  Vol.   14  (3),  p.   197. 

SCHULTZE. 

Ann.  d.  Physik.,  1801,  Bd.  V.,  p.  140. 


59 


SCHUMACHER. 

Astronomische  Nachtrichten,  Bd.  40,  1855. 
SCHUMANN. 

Wied.  Ann.,  1884,  Bd.  23,  p.  353. 
STEFAN. 

Wien.  Ber.  Mathem.  Naturw.,  1862  (2),  Vol.  46,  pp.  8,  495. 

1872  (2),  Vol.  65,  p.  360. 
STEUDEL. 

Wien.  Ann.,  1882,  Bd.  16,  p.  369. 
STEWART  AND  TAIT. 

Proc.  Royal  Society,  1865,  Vol.  14,  p.  339. 

Phil.  Mag.  (4),  Vol.  30,  p.  314. 
STOKES. 

Camb.  Phil.  Trans.,  1849,  Vol.  8,  p.  287. 
1850,  Vol.  9,  p.  8. 

Phil.  Mag.,  1851  (4),  Vol.  1,  p.  337. 
ST.  VENANT,  BARRE  DE. 

Compt.  Rend.,  17,  1843,  pp.  1140  and  1240. 
SUTHERLAND. 

Proc.  Ann.,  Acad.  1885,  p.  13. 

Phil.  Mag.,  1893  (5),  Vol.  36,  p.  507. 

TOMLINSON. 

Phil.  Trans.,  1886,  Vol.  177  (2),  p.  767. 
WARBURG. 

Pogg.  Ann.,   1876,  Bd.   159,  p.  399. 
WARBURG  AND  BABO. 

Wied.  Ann.,  1882,  Bd.  17,  p.  390. 

Sitz.  d.  Berl.  Akad.,  1882,  p.  509. 

WlEDEMANN,   G. 

Pogg.  Ann.,  1856,  Bd.  99,  p.  177. 

WlEDEMANN,   E. 

Arch.  d.  Sc.  Phys.  et  Nat.  de  Geneve,  1876,  Vol.  56,  p.  277. 
Fortschr.  d.  Phys.,  1876,  Vol.  32,  p.  206. 


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